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Macroscopic Balances, Summaries of Chemistry

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.

Typology: Summaries

2022/2023

Uploaded on 10/08/2022

donkeydone
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Topic 4 - Macroscopic Balances
consider the following system
mass balance
-
mole balance
-
energy balance
-
momentum balance
-
intrinsic properties are inherent to the fluid:
extrinsic properties are dependent on quantity:
how can you decide? ask what happens when you
cut the amount of material in half.
energy (extrinsic):
energy/mass (intrinsic):
.
4.A Mass Balance (Text 4.1)
4.B Energy Balance (Text 4.2A-E)
4.C Mechanical Energy Balance (Text 4.2F)
4.D Bernoulli Equation (Text 4.2G)
4.E Linear Momentum Balance (Text 4.3)
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Topic 4 - Macroscopic Balances

• consider the following system

- mass balance

- mole balance

- energy balance

- momentum balance

• intrinsic/extrinsic properties

intrinsic properties are inherent to the fluid:

extrinsic properties are dependent on quantity:

how can you decide? ask what happens when you

cut the amount of material in half.

energy (extrinsic):

energy/mass (intrinsic):

4.A – Mass Balance (Text 4.1)

4.B – Energy Balance (Text 4.2A-E)

4.C – Mechanical Energy Balance (Text 4.2F)

4.D – Bernoulli Equation (Text 4.2G)

4.E – Linear Momentum Balance (Text 4.3)

4.A - Mass Balance (Text 4.1)

  1. Unsteady-State Mass Balance, Part 1: the easy way
    • consider the following system, a nozzle
    • mass balance on control volume:
    • if we assume steady-steady state:
  • if we also assume the fluid is incompressible (okay for liquids):

Because fluid velocity, v, and density, ρ, can vary with position, the mass flux across an interface, Ai, is

  • when only one inlet and one outlet

3.Unsteady-State Species Mass Balance

  • If we have multiple species within a fluid:
  • define mass fraction, wi:
  • Can write an unsteady-state mass on specie i:

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

  1. Components of an energy balance
  2. Adding energy to control volume .
  3. Consider the material within the control volume: Your tasks: 1.Learn what all the terms are 2. Learn the units for each term Learn the signs for each term (why some are positive and others are negative) 3.

control surface 2: exit

  1. Kinetic energy velocity correction Note that control surface 3: wall assume over small surface that H 1 , z 1 , ρ 1 constant, but v 1 varies (e.g., no slip boundary condition at pipe wall) assuming steady-state: writing out the energy balance: energy balance: mass balance: rearranging, and dividing by ṁ

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

  • Assumes uniform H, z, ρ, T, P for each inlet/outlet
  • H = f (T,P) (enthalpy per unit mass)

at 1: at 2: Recap: Consider a simple system with one inlet and one outlet

  • Unsteady-state energy balance:
  • Steady state:

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:

or

When we plug this into our steady-state energy balance

we get if we assume fluid is incompressible

  1. Text Example 4.2- 3. A pump is used to deliver water. then Note: • Mechanical energy balance includes shaft work (turbine/pump) - Includes viscous losses (J/kg or lbf-ft/lbm) - Does not explicitly account for heating/cooling: Q Frictional losses, Fi, involve conversion of mechanical work to thermal energy (term must always be positive)

For simple case of pumping a fluid around in circles:

a)what is the shaft work per kilogram (-Ws) assuming no frictional losses?

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?

c) assuming that the pump is 65% efficient, how much work (in J/kg) is needed?

d) the system operates at 5000 L/h. What is the power consumption?

Recap: we have two energy balances that are equivalent

  • One includes thermal energy terms: H, Q

4.D – Bernoulli Equation (Text 4.2G)

Recall the mechanical energy balance from Topic 4.C

  • if no
  • and no
  • and this can also be combined with a mass balance:
  1. Example: Estimating flow rate through a pipe If we measure P 1 and P 2 , can we estimate v 2? assumptions:

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?

  1. Example: Draining of a Tank What is velocity of fluid leaving tank, v 2? What is the mass flow rate, ṁ 2? How long does it take to drain the tank, tdrain? assumptions: