THE BERNOULLI EQUATION: LIMITATIONS AND APPLICATIONS [ physical interpretation: what are we doing today? ] The Bernoulli Equation empowers us through a simplified energy balance to solve a variety of fluids problems where we know energy conditions at one site and wish to know them at another The Bernoulli Equation can be derived from Newton’s Second Law when it is considered relative to a streamline The “BE” can also be had from an adaptation of the general Energy Equation Who Cares!? from a simplified fluid dynamics view, the BE will really allow us to get quick and reasonable assessments of the energy balance between two points, again like the energy equation the applications are nearly too vast to mention 87-351 Fluid Mechanics and Hydraulics

THE ENERGY EQUATION [ derivation ] The energy equation is really a statement of the first law of thermodynamics when this law is applied to a system we can write or, expressing this mathematically, - (1) or, simply - (2) 87-351 Fluid Mechanics and Hydraulics

THE ENERGY EQUATION [ derivation cont’d ] For now, we maintain the generality of the expression, and thus for our understanding will define some of the parameters expressed in (2) we see that e is defined by - (3) where uhat represents internal energy per unit mass, V2/2 is the kinetic energy per unit mass, and finally potential energy per unit mass is represented by gz - (2) Qdotnetin denotes the net rate of heat transfer into the system, and Wdotnetin reps the net rate of work transfer, +ve transfer is “in” to the system and –ve transfer is “out” of the system 87-351 Fluid Mechanics and Hydraulics

THE ENERGY EQUATION [ derivation cont’d ] For our convenience, we will adapt (2) to be expressed in a control volume representation so, thus for a CV that is coincident with the system at an instant in time - (4) and for a fixed, and non-deforming CV we can write (and by now, we all knowthis is RTT (yet again)) - (5) here the intrinsic parameter, b, is replaced by e, so we can say - (6) 87-351 Fluid Mechanics and Hydraulics

THE ENERGY EQUATION: APPLICATION [ example 1 ] GIVEN: (adiabatic pumping process) REQD: Determine the power required by the pump 87-351 Fluid Mechanics and Hydraulics

THE ENERGY EQUATION: APPLICATION [ example 1 (cont’d) ] SOLU: We construct our CV to include the flow entrance and exit sections (adiabatic pumping process) Let’s apply a specific form of the energy eqn that is suited for one-dimensional flow that is steady in the mean - (1) 87-351 Fluid Mechanics and Hydraulics

THE ENERGY EQUATION: APPLICATION [ example 1 (cont’d) ] SOLU: So, we examine (1) and attempt to acquire each parameter - (1) we know from the given data that the rise in internal energy of the water, (uout-uin) = 3000ft∙lb/slug, (this is associated with the temperature rise across the pump) (adiabatic pumping process) we can also make some simplifications to (1) - (2) 87-351 Fluid Mechanics and Hydraulics

THE ENERGY EQUATION: APPLICATION [ example 1 (cont’d) ] SOLU: We can easily solve the rest from our knowledge of the conservation of mass mdot is easily had, and the velocities follow simply (adiabatic pumping process) - (3) - (4) - (5) 87-351 Fluid Mechanics and Hydraulics

THE ENERGY EQUATION: APPLICATION [ example 1 (cont’d) ] SOLU: Plugging all the numbers in now - (ans) it is interesting to consider, when we look at this expression, that 22% of the total power required is demanded by the internal energy change (temperature rise), and 23% is required to raise the pressure, and finally the kinetic energy increase demands the remaining 55% of energy input 87-351 Fluid Mechanics and Hydraulics

THE BERNOULLI EQUATION: LIMITATIONS AND APPLICATIONS [ a form of the energy equation ] If we consider the form of the Energy Equation that has been adapted for a one dimensional, steady in the mean flow, for a flow that is incompressible, we have - (1) If we re-arrange terms and divide by mass flux, mdot, we have - (2) where 87-351 Fluid Mechanics and Hydraulics

THE BERNOULLI EQUATION: LIMITATIONS AND APPLICATIONS [ a form of the energy equation (cont’d) ] If the flow we are considering also has negligible friction effects (viscosity) then the Bernoulli Equation (derived from F=ma) can be used in this vein to describe what happens between two sections in the flow - (3) Let us divide (3) by density,r - (4) 87-351 Fluid Mechanics and Hydraulics

THE BERNOULLI EQUATION: LIMITATIONS AND APPLICATIONS [ a form of the energy equation (cont’d) ] It is obvious to us by comparing (4) and (2), that the additional assumption of an inviscid fluid, must imply that - (5) and of course when there is friction - (6) We can really refer to terms in (5) and (6) as “energy loss” terms and therefore the combination of - (7) might be considered as an expression of “available” or “useful” energy 87-351 Fluid Mechanics and Hydraulics

THE BERNOULLI EQUATION: LIMITATIONS AND APPLICATIONS [ a form of the energy equation (cont’d) ] This said, we can re-express (2) as a Bernoulli form of the Energy Equation that accounts for losses experienced by the flow - (8) Civil Engineers often like to express the Bernoulli Equation in terms of “head”, or energy per unit weight - (9) loss term 87-351 Fluid Mechanics and Hydraulics

THE BERNOULLI EQUATION: LIMITATIONS AND APPLICATIONS [ example 1 ] GIVEN: Qpump = 2 cfs Pump adds 10 hp REQD: Determine the head loss and power loss associated with this flow 87-351 Fluid Mechanics and Hydraulics

THE BERNOULLI EQUATION: LIMITATIONS AND APPLICATIONS [ example 1 (cont’d) ] SOLU: 1. Let us write the Bernoulli Equation for this flow - (E1) If we select the surfaces of each lake as our control points (A) and (B), we lose the pressure terms, and the velocity terms in (E1) we are left with - (E2) here, hL represents head loss, and hs represents head supplied and we know the pump head can be obtained from - (E3) 87-351 Fluid Mechanics and Hydraulics

THE BERNOULLI EQUATION: LIMITATIONS AND APPLICATIONS [ example 1 (cont’d) ] SOLU: so we write - (E3) Then we solve for head loss from (E2) - (ans) Further, we can obtain the power loss due to friction through the following - (E4) 87-351 Fluid Mechanics and Hydraulics

THE BERNOULLI EQUATION: LIMITATIONS AND APPLICATIONS [ example 1 (cont’d) ] SOLU: - (E4) - (ans) we observe that remaining 6.80 hp, (10 hp – 3.20 hp), is power used by the pump to lift the water through the 30’ head 87-351 Fluid Mechanics and Hydraulics

THE BERNOULLI EQUATION: LIMITATIONS AND APPLICATIONS [ discussion of assumptions and limitations ] 1. In the derivation of the Bernoulli Equation some pretty significant assumptions are made it is assumed that the flow is INCOMPRESSIBLE STEADY IRROTATIONAL INVISCID for valid results to be realized from the equation 87-351 Fluid Mechanics and Hydraulics

THE BERNOULLI EQUATION: LIMITATIONS AND APPLICATIONS [ discussion of assumptions and limitations: compressibility effects ] Compressibility incompressibility is a reasonable assumption for most liquids but can be an issue in gases stems from the fact that the stagnation pressure is greater than the static pressure by an amount rV2/2 (the dynamic pressure) as the dynamic pressure gets larger compared to the static pressure, the density change between two points can become prohibitively large (as the dynamic pressure varies as V2) if we like, we can adapt the Bernoulli Equation for compressibility effects with the addition of the term - (10) 87-351 Fluid Mechanics and Hydraulics

THE BERNOULLI EQUATION: LIMITATIONS AND APPLICATIONS [ discussion of assumptions and limitations: unsteady effects ] Unsteady Effects The BE was obtained by integrating the component of Newton’s second law along a streamline when we perform the integration, the integration of the acceleration term 0.5rd(V2) gives rise to the kinetic energy term in the BE if unsteady effects are to be accounted for the ∂V/∂t term must be included the integration of this term is difficult because we have to know something about the variation of ∂V/∂t along the streamline (generally not known) 87-351 Fluid Mechanics and Hydraulics

THE BERNOULLI EQUATION: LIMITATIONS AND APPLICATIONS [ discussion of assumptions and limitations: rotational effects ] Rotational Effects The BE cannot be applied across streamlines we know the Bernoulli constant as - (11) in general, this constant varies from streamline to streamline which prevents us from applying the equation across them 87-351 Fluid Mechanics and Hydraulics

THE BERNOULLI EQUATION: LIMITATIONS AND APPLICATIONS [ applications: example 1 ] GIVEN: assume inviscid REQD: Determine the flow rate and the pressure at (2) 87-351 Fluid Mechanics and Hydraulics

THE BERNOULLI EQUATION: LIMITATIONS AND APPLICATIONS [ applications: example 1 (cont’d) ] SOLU: 1. Let us write the Bernoulli Equation for this flow we first learn about (1) before we solve for the parameters at (2) - (E1) use continuity to express V1 in terms of V3 - (E2) then plug what we know into (E1) and solve for V3 - (E3) 87-351 Fluid Mechanics and Hydraulics

THE BERNOULLI EQUATION: LIMITATIONS AND APPLICATIONS [ applications: example 1 (cont’d) ] SOLU: Now we can pick up the flow rate and examine the energy balance between (1) and (2) get Q - (ans) write BE between (1) and (2) - (E4) which yields - (ans) 87-351 Fluid Mechanics and Hydraulics

THE BERNOULLI EQUATION: LIMITATIONS AND APPLICATIONS [ applications: example 2 ] GIVEN: assume inviscid REQD: At what height, h, will cavitation occur? Then, to remedy cavitation, would we increase or decrease the value of D1 and/or D2? 87-351 Fluid Mechanics and Hydraulics

THE BERNOULLI EQUATION: LIMITATIONS AND APPLICATIONS [ applications: example 2 ] SOLU: 1. Let us write the Bernoulli Equation for this flow (from (0) to (1)) assume inviscid - (E1) so - (E2) but, we know - (E3) 87-351 Fluid Mechanics and Hydraulics

THE BERNOULLI EQUATION: LIMITATIONS AND APPLICATIONS [ applications: example 2 (cont’d) ] SOLU: Now, of course we can write BE from (0) to (2) assume inviscid - (E4) and, we know - (E5) we get then - (E6) - (E7) 87-351 Fluid Mechanics and Hydraulics

THE BERNOULLI EQUATION: LIMITATIONS AND APPLICATIONS [ applications: example 2 (cont’d) ] SOLU: Let us then combine (E2), (E6), and (E7) - (E6) - (E2) assume inviscid - (E7) - (E8) which we may rewrite as - (E9) dumping the numbers in we get - (ans) 87-351 Fluid Mechanics and Hydraulics

THE BERNOULLI EQUATION: LIMITATIONS AND APPLICATIONS [ applications: example 2 (cont’d) ] SOLU: To answer the second question, regarding the influence the size D1 and D2 have on the cavitation, we re-examine (E9) assume inviscid - (E9) with inspection of (E9), this expression tells us a decrease in D2 will increase the height we can fill to without cavitation, and an increase in D1 will increase the height we can fill to without cavitating 87-351 Fluid Mechanics and Hydraulics