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The various macroscopic balances, including mass balance, mole balance, energy balance, and momentum balance, for a given system. It covers the concepts of intrinsic and extrinsic properties, and how to determine them. The document then delves into the unsteady-state mass balance, energy balance, and mechanical energy balance, providing examples and mathematical expressions to solve these problems. It also introduces the bernoulli equation and its application in estimating flow rates. Finally, the document explores the linear momentum of a control volume and how to construct an unsteady-state momentum balance. The content covers a wide range of topics related to macroscopic balances, making it a comprehensive resource for students studying fluid mechanics, thermodynamics, or related engineering disciplines.
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4.A - Mass Balance (Text 4.1)
Because fluid velocity, v, and density, ρ, can vary with position, the mass flux across an interface, Ai, is
3.Unsteady-State Species Mass Balance
Example: A tank initially (t=0) holds 500 kg of a 10% solution of salt (wA=0.1). It is fed a 20% solution of salt at 10 kg/h, and the solution from the tank is drained at a rate of 5 kg/h. Assume the tank is well-mixed. Develop an algebraic expression for the mass of solution in the tank as a function of time, m(t). a) b) Develop an algebraic expression for the concentration of the salt solution (notice that α remains constant for each surface, so can pull it out of each surface integral) α < 90 α > 90 α = 90 We can break our integral over control surface into the sum of 3 integrals: And if we assume steady state and an incompressible fluid:
Develop an algebraic expression for the concentration of the salt solution as a function of time, wA(t). b) a) mass balance on tank: b) salt balance on tank: apply product rule: rearranging: note that at t=0, wA=0.1. this defines lower limit of integration
recap: Can construct a simple unsteady-state mass balance: we don't need vectors for simple systems with a few inlets and outlets: can apply this to an individual specie: To solve mass balance problems, we can compose these equations for our system and then solve them. if density is a constant if density constant
4.B - Energy Balance (Text 4.2A-B) define total energy of this material as the sum of internal energy, kinetic energy, and potential energy these are extrinsic, because alternatively, we can make them intrinsic: so energy of material in control volume rate of accumulation of energy rate of energy efflux by flow rate of heat energy added rate of shaft work energy removed
control surface 2: exit
unless velocity is uniform (i.e., plug flow) However, for laminar (parabolic) flow through a circular pipe of radius R: and so the book shows that: hence, we can define the ratio as α going back to energy balance, we can incorporate α Notes: • One inlet (1), one outlet (2), steady-state: |ṁ 1 | = |ṁ 2 |= ṁ
at 1: at 2: Recap: Consider a simple system with one inlet and one outlet
Do not memorize this equation. To solve an energy balance problem, start with a generic balance and modify it as needed (adding/removing terms)
So that:
we get if we assume fluid is incompressible
For simple case of pumping a fluid around in circles:
additional information: fluid is water, ρ = 998 kg/m^3 inlet/outlet pipes are same diameter Reynolds number > 4000 b) if the real shaft work is Ws = - 155.4 J/kg, what are the frictional losses?
Recap: we have two energy balances that are equivalent
Recall the mechanical energy balance from Topic 4.C
combine 2 equations:
→ Venturi meter then get: Bernoulli Equation ① - inlet ② - outlet
so that if water flows from a 5 cm diameter pipe into a 2.5 cm diameter pipe, and the measured pressure drop is P 2 - P 1 = 0.1 bar, what is v 2? laminar or turbulent flow in smaller pipe?