SYSTEMS VS. CONTROL VOLUMES System (closed system): Predefined mass m, surrounded by a system boundary Control volume CV (open system): Arbitrary geometric space, surrounded by control surfaces (CS) Many physical laws are formulated initially for closed systems 1. The mass of a system is constant. Mass balance: dm dt = 0. The time derivative of the linear momentum for a system equals the sum of forces acting on the system. Momentum balance: d dt (mv) = F 3. The time derivative of the total energy E for a system equals the net transport of heat and work interaction per unit time. Energy balance: de dt = d dt (me) = Q net,in Ẇnet,out = Q Ẇ How convert these relations to control volumes? Ch. 3.1 Fluid Mechanics C. Norberg, LTH
THE REYNOLDS TRANSPORT THEOREM Let B be an arbitrary mass-dependent quantity; b is the same quantity expressed per unit mass. Also let a system at an arbitrary time t coincide with a control volume CV. The Reynolds theorem then is: db dt = t B CV +Ḃnet,out = t bρdv + bρ(v n)da CV CS The unit vector n points out from the local control surface. Mass and quantity B is transported over the surface through the normal velocity component, V n = V n. Moving but non-deforming control volumes: : db dt = CV t (ρb)dv + CS bρ(v r n)da where V r is the fluid velocity relative to a coordinate system that is fixed to the control volume. In the following, V = V r (nondeforming control volumes). Ch. 3. Fluid Mechanics C. Norberg, LTH
CONSERVATION OF MASS Reynolds transport theorem: db dt = t B CV +Ḃnet,out B = m, b = 1; mass balance dm dt = m CV t Mass flow rate: ṁ = ρ(v n)da [kg/s] A +ṁ net,out = 0 Volume flow rate: Q = A (V n)da [m3 /s] Average velocity: V = Q/A = A 1 (V n)da [m/s] A If density variations over the surface are negligible, ṁ = ρva. Stationary conditions, many inlets and outlets, mass balance: ṁ out = ṁ in Incompressible flow, ρ = const. Q out = Q in,or (VA) out = (VA) in Ch. 3.3 Fluid Mechanics C. Norberg, LTH
THE LINEAR MOMENTUM EQUATION d Newton: dt (mv) = F sys V is the velocity vector relative to a non-accelerating (inertial) coordinate system Forces act momentarily without memory effects F sys = F CV Consider now a non-deforming CV that moves with a constant velocity relative to an inertial coordinate system. A coordinate system (xyz) fixed to the CV then also is an inertial system The Reynolds transport theorem with b = V gives CV t (ρv)dv + Vρ(V n)da = F CS CV where V is the fluid velocity vector relative to xyz. Stationary flow, homogeneous conditions over inlets and outlets: (ṁv) out (ṁv) in = F CV Same as above but only one inlet and one outlet (ṁ out = ṁ in = ṁ): ṁ(v out V in ) = F CV Correction for velocity variations over inlets and outlets: (βṁv) out (βṁv) in = F CV, β = A 1 (u/v) da Fully developed laminar pipe flow: u/u 0 = 1 (r/r) V/U 0 = 1/, β = 4/3 Fully developed turbulent pipe flow: u/u 0 (1 r/r) m V/U 0 (1+m) 1 (+m) 1 β (1+m)(+m) (1+m) 1 /4 m = 1/7 u/u 0 0.8, β 1.0 Ch. 3.4 Fluid Mechanics C. Norberg, LTH
THE BERNOULLI EQUATION For frictionless flow along a streamline or along pipe sections with negligible cross-sectional variations and wall friction, the Bernoulli equation reads (stationary incompressible flow): 1 p+ ρv +ρgz = konst. In particular for horizontal flow or negligible gravitational effects: p+ ρv = konst. If velocity increases the pressure decreases, and vice versa. Confining liquid flow; volume flow rate, Q = VA = const. decreasing area velocity increases Bernoulli equation pressure decreases Flow around a NACA 441 airfoil, angle-of-attack 5 Crowded streamlines on upper frontal side higher velocity, lower pressure upward force (lift) 1 Daniel Bernoulli, 1700 178, Holland/Switzerland. Ch. 3.5 Fluid Mechanics C. Norberg, LTH
THE PRANDTL (PITOT-STATIC) TUBE Stagnation point at the frontal pressure tap, pressure p s ; at the pressuretaportapssomewhatdownstreamalongthetubeandbycareful design the pressure has recovered exactly to the static pressure of the oncoming stream p. Along the oncoming horizontal streamline ending at the stagnation point the retarded flow can be regarded as frictionless. From Bernoulli equation, p s = p +ρu /, i.e., U = (p s p)/ρ.measuredp s pandknownfluiddensityρ U. Yaw angles within ±0 error in U within 1% Frictional effects are negligible if Re > 100 (approx.), when based on the tube outer diameter d. Ch. 3.5 (6.1) Fluid Mechanics C. Norberg, LTH
THE EXTENDED BERNOULLI EQUATION Energy balance, stationary conditions, control-volume approach: CS eρ(v n)da = Q Ẇ Energy per unit mass, e = û + V / + gz; internal energy û, z upwards. Work interactions per unit time (power), Ẇ = Ẇs+Ẇv+ Ẇ p ; shaft power Ẇ s, viscous power Ẇ v, Ẇ p = CSp(V n)da. Enthalpy per unit mass, ĥ = û+p/ρ CS ( ĥ+v /+gz ) ρ(v n)da = Q Ẇs Ẇv For incompressible flow in pipe and ducts the viscous work interactions are negligible; homogenous inlet and outlet (ṁθ) out (ṁθ) in = Q Ẇs, θ = ĥ+v /+gz Some rearrangements gives the extended Bernoulli equation: ( p+ ρv ( +ρgz )in = p+ ρv ) +ρgz +ρw s + p f out where w s = Ẇs/ṁ; p f > 0 is the irreversible pressure loss. 3 An alternative form, eq. (3.73), comes from the division of γ = ρg: p γ + V g +z in = p γ + V g +z out h pump +h turbine +h f Head loss, h f = p f /γ; pump head, h pump ; turbine head, h turbine. Correction for non-uniform cross-sectional velocity variations: ρv / αρv /, where α = A 1 A(u/V) 3 da; fully developed pipe flow: α = ; turbulent, m = 1/7 α = 1.06. 3 That p f > 0 follows from the second law of thermodynamics. Ch. 3.7 Fluid Mechanics C. Norberg, LTH