Basic Laws for a System

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Basic Laws for a System

• The basic laws are:

– Conservation of mass, – Newton’s second law,

– angular-momentum principle, – first law of thermodynamics.

rate equations

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Basic Laws for a System

• For a system (a fixed amount of matter, M):

Conservation of mass

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sum of all external forces acting on the system =time rate of change of linear momentum of the system,

Newton’s Second Law

دنیآرب یاهورین

متسیس کی رب دراو یجراخ =متسیس یطخ موتنموم تارییغت خرن

𝐹 = 𝑚 Ԧ𝑎 → ԦԦ 𝐹 = 𝑚𝑑𝑉

𝑑𝑡 = 𝑑(𝑚𝑉)

𝑑𝑡 = 𝑑𝑃 𝑑𝑡

𝐿𝑖𝑛𝑒𝑎𝑟 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚 𝑃 = 𝑚𝑉

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Basic Laws for a System

Newton’s Second Law

𝑑𝑚 & 𝑑𝑉

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The Angular-Momentum Principle

Angular 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚 H = 𝑚 Ԧ𝑟 × 𝑉

𝑃 = 𝑚𝑉 → 𝑚𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝑙𝑖𝑛𝑒𝑎𝑟 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚: Ԧ𝑟 × 𝑚𝑉 = 𝐻

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rate of change of angular momentum is equal to the sum of all torques acting on the system

دنیآرب یاهرواتشگ

متسیس کی رب دراو یجراخ =متسیس یا هیواز موتنموم تارییغت خرن

(8)

Basic Laws for a System

• Torque can be produced by surface and body forces (here gravity) and also by shafts that cross the system boundary,

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• conservation of energy for a system:

• ሶ𝑄 (the rate of heat transfer) is positive when heat is added to the system from the surroundings;

• 𝑊ሶ (the rate of work) is positive when work is done by the system on its surroundings.

The First Law of Thermodynamics

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Basic Laws for a System

The First Law of Thermodynamics

• u is the specific internal energy, V the speed, and z the height (relative to a convenient datum) of a particle of substance having mass dm.

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Reynolds Transport Theorem (RTT)

• For a system of fluid,

Relation of System Derivatives to the Control Volume Formulation: RTT

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Reynolds Transport Theorem (RTT)

• Gauss's theorem (سناژروید یروئت⁾

Reynolds Transport Theorem

𝑭

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Gauss's theorem

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Remember from chapter 1?

𝑑𝜂

𝑑𝑡 = 𝜕𝜂

𝜕𝑡 + 𝑢𝜕𝜂

𝜕𝑥 + 𝑣𝜕𝜂

𝜕𝑦 + 𝑤𝜕𝜂

𝜕𝑧

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خرن تارییغت لک

متسیس رد N لرتنک مجح رد N تارییغت خرن نامز رد

تارییغت خرن لرتنک مجح رد N

حوطس زا جورخ و دورو هجیتن رد لرتنک

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inlet outlet

Temporal

Local Local

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Conservation of Mass

• mass conservation principle: The mass of the system remains constant 𝑑𝑀

𝑑𝑡 )_{𝑠𝑦𝑠𝑡𝑒𝑚} = 0

1 1

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Conservation of Mass

• 1^st term represents the rate of change of mass within C.V.

• 2^nd term represents the net rate of mass flux out through C.V.

The mass conservation equation is also called the Continuity equation

یگتسویپ هلداعم :

مرج یاقب لصا

ሶ

𝑚 = න

𝐶𝑆𝜌𝑉. 𝑛 𝑑𝐴 : 𝑀𝑎𝑠𝑠 𝑓𝑙𝑢𝑥

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Conservation of Mass: Special Cases

• For an incompressible fluid,

• conservation of mass for incompressible flow through a fixed (non- deformable) control volume:

න

𝐶𝑆𝑉. 𝑛 𝑑𝐴 = 0 𝑄 = න

𝐶𝑆

𝑉. 𝑛 𝑑𝐴

یبد یاقب

؟؟یمجح

!!

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𝑉 ₁

𝑉 ₃ 𝑉 ₂

𝑉 ₄

𝑐𝑜𝑛𝑡𝑟𝑜𝑙 𝑣𝑜𝑙𝑢𝑚𝑒

−𝑉₁𝐴₁ − 𝑉₂𝐴₂ + 𝑉₃𝐴₃ + 𝑉₄𝐴₄ = 0

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Conservation of Mass

• uniform velocity at each inlet and exit:

• At steady state+incompressible:

න

𝐶𝑆

𝑉. 𝑛 𝑑𝐴 = 0

෍𝐶𝑆𝑉. Ԧ𝐴 = 0

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• During a steady-flow process, the total amount of mass contained within a control volume does not change with time.

Mass Balance for Steady-Flow Processes

𝒎_𝑪𝑽 = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕

෍𝒊𝒏𝒎 = ෍ሶ

𝒐𝒖𝒕𝒎ሶ (𝒌𝒈 𝒔 )

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Conservation of Mass

• Many engineering devices such as nozzles, diffusers, turbines, compressors, and pumps involve a single stream (only one inlet and one outlet)

single-stream steady-flow systems

ሶ

𝒎_𝟏 = ሶ𝒎_𝟐 → 𝝆_𝟏𝑽_𝟏𝑨_𝟏 = 𝝆_𝟐𝑽_𝟐𝑨_𝟐

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• The conservation of mass relations can be simplified even further when the fluid is incompressible, which is usually the case for liquids.

Incompressible Flow

𝑺𝒕𝒆𝒂𝒅𝒚 𝒊𝒏𝒄𝒐𝒎𝒑𝒓𝒆𝒔𝒔𝒊𝒃𝒍𝒆 𝒇𝒍𝒐𝒘: ෍

𝒊𝒏 ሶ𝑸 = ෍

𝒐𝒖𝒕 ሶ𝑸 (𝒎^𝟑 𝒔 )

Steady incompressible flow (single-stream) ሶ𝑸_𝟏 = ሶ𝑸_𝟐 → 𝑽_𝟏𝑨_𝟏 = 𝑽_𝟐𝑨_𝟐

Conservation of Volume Principle

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Conservation of Mass

𝑁𝑜𝑟𝑚𝑎𝑙 𝑐𝑜𝑚𝑝𝑜𝑛𝑒𝑛𝑡 𝑜𝑓 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦: 𝑉_𝑛 = 𝑉 cos 𝜃 = 𝑉. 𝑛

Inlet: - Outlet: +

𝜕𝑚_𝐶𝑉

𝜕𝑡 + ෍

𝑜𝑢𝑡

ሶ

𝑚 − ෍

𝑖𝑛

ሶ

𝑚 = 0

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෍𝐶𝑆𝑉. Ԧ𝐴 = 0

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𝜕𝑚_𝐶𝑉

𝜕𝑡 + ෍

𝑜𝑢𝑡

ሶ

𝑚 − ෍

𝑖𝑛

ሶ

𝑚 = 0

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Moving Control Volumes

𝑽_𝒓 = 𝑽 − 𝑽_𝑪𝑽

fluid velocity C. V. velocity

Relative velocity: یبسن تعرس

𝑽_𝒓 → 𝑽

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Deforming Control Volumes

𝑽_𝒓 = 𝑽 − 𝑽_𝑪𝑺

fluid velocity Control Surface velocity:

لرتنک حطس یاهزرم ییاج هباج تعرس

𝑽_𝒓 → 𝑽

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Conservation of Mass

ሶ

𝑚_𝑏𝑐? ? ? ?

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Linear Momentum Equation

Ԧ𝐹 = 𝑚 Ԧ𝑎 → Ԧ𝐹 = 𝑚𝑑𝑉

𝑑𝑡 = 𝑑(𝑚𝑉)

𝑑𝑡 = 𝑑𝑃 𝑑𝑡

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Reynolds Transport Theorem

𝑉 = (𝑢, 𝑣, 𝑤)

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• Uniform flow at each inlet and exit:

خرن تارییغت لک

موتنموم متسیس رد یطخ

=

متسیس رب دراو یجراخ یاهورین

تارییغت خرن موتنموم

د یطخ ر

نامز رد لرتنک مجح

تارییغت خرن موتنموم

د یطخ ر

و دورو هجیتن رد لرتنک مجح لرتنک حوطس زا جورخ

(43)

Momentum Equation

• three scalar components of momentum equation:

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• three scalar components of momentum equation:

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Momentum Equation

• At steady state:

• steady state+uniform flow:

෍ Ԧ𝐹 = න

𝐶𝑆

𝑉𝜌 (𝑉. 𝑛) 𝑑𝐴

෍ Ԧ𝐹 = ෍

𝐶𝑆𝑉𝜌 (𝑉. 𝑛) 𝐴

(46)

෍ Ԧ𝐹 = ෍

𝑜𝑢𝑡𝑉𝜌 (𝑉. 𝑛) 𝐴 − ෍

𝑖𝑛𝑉𝜌 (𝑉. 𝑛) 𝐴

ሶ𝑚 = ׬_𝐶𝑆𝜌𝑉.𝑛 𝑑𝐴 یمرج یبد

෍𝐶𝑆𝑉𝜌 (𝑉. 𝑛) 𝐴 = 𝑚𝑉ሶ _𝑜𝑢𝑡 − 𝑚𝑉ሶ _𝑖𝑛

(47)

ሶ

𝑚𝑉 ₁

ሶ

𝑚𝑉 ₃ 𝑚𝑉ሶ ₂

ሶ

𝑚𝑉 ₄

𝑐𝑜𝑛𝑡𝑟𝑜𝑙 𝑣𝑜𝑙𝑢𝑚𝑒

(48)

• Unfortunately, the velocity across most inlets and outlets of practical engineering interest is not uniform. Nevertheless, it turns out that we can still convert the control surface integral into algebraic form, but a dimensionless correction factor, called the momentum-flux correction factor, is required,

Momentum-Flux Correction Factor

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Linear Momentum Equation

• For the case in which density is uniform over the inlet or outlet and V is in the same direction as V_avg over the inlet or outlet,

𝜷 ≥ 𝟏

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Laminar

Turbulent

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• the net force acting on the control volume during steady flow is equal to the difference between the rates of outgoing and incoming momentum flows.

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Linear Momentum Equation

• single-stream systems:

Steady Flow with One Inlet and One Outlet

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• V: average velocities across the inlet and outlet

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Governing equations

Mass conservation:

Momentum conservation:

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Momentum Equation

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Note that the choice of CVII meant we needed an additional free-body diagram. In general it is best to select the control volume so that the force

sought acts explicitly on the control volume.

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Momentum Equation

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Governing equations

Mass conservation???

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𝑉₁????

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𝑉₁????

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• subtract the atmospheric pressure and work with gage pressures

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Momentum Equation

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ار ه ص مع .دراد ایرج ه ص کی یور ایاپ یار رد ری پان مکارت ایرج کی تعرس لی ورپ .دیریگب ر ن رد 𝒃

طقم حطس رد لایس و 𝒂𝒃

:دنک یم تیعبت ریز یاه هطبار زا 𝒄𝒅

@𝒙 = 𝒙_𝒂:𝒖 = 𝑼 & @𝒙 = 𝒙_𝒅: 𝒖

𝑼 = 𝟐 𝒚

𝜹 − 𝒚 𝜹

𝟐

𝜹 و رد لایس هی تما 𝒙 = 𝒙_𝒅

رگا .تسا 𝒙_𝒅 − 𝒙_𝒂 = 𝑳

تسبو طم :

.a حطس زا لایس یمرج یبد .𝒃𝒄

.b زرم رد لایس سوتم تعرس .𝒃𝒄

.c زرم رد ه ص رب دراو یورین .لایس ر زا 𝒂𝒅

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Control Volume Moving with Constant Velocity

• two coordinate systems: XYZ, “absolute,” or stationary coordinates, and the xyz coordinates attached to the control volume

• velocities must be measured relative to the control volume. (It is helpful to imagine that the velocities are those that would be seen by an observer moving with the control volume.)

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Control Volume Moving with Constant Velocity

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Momentum Equation for Control Volume with Rectilinear Acceleration

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Momentum Equation for Control Volume with Rectilinear Acceleration

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𝑈𝑠𝑒 𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑖𝑒𝑠

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Momentum Equation for Control Volume with Rectilinear Acceleration

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یروآدای

Angular 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚 H = 𝑚 Ԧ𝑟 × 𝑉

𝑃 = 𝑚𝑉 → 𝑚𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝑙𝑖𝑛𝑒𝑎𝑟 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚: Ԧ𝑟 × 𝑚𝑉 = 𝐻

(100)

یروآدای

rate of change of angular momentum is equal to the sum of all torques acting on the system

دنیآرب یاهرواتشگ

متسیس کی رب دراو یجراخ =متسیس یا هیواز موتنموم تارییغت خرن

(101)

یروآدای

• Torque can be produced by surface and body forces (here gravity) and also by shafts that cross the system boundary,

(102)

The Angular-Momentum Principle

(103)

The Angular-Momentum Principle: Special cases

Steady state

𝑇_{𝑠𝑦𝑠𝑡𝑒𝑚} = ෍ 𝑀 = න

𝐶𝑆

Ԧ𝑟 × 𝑉 𝜌 𝑉. 𝑛 𝑑𝐴

(104)

The Angular-Momentum Principle: Special cases

Steady state+uniform flow

𝑇_{𝑠𝑦𝑠𝑡𝑒𝑚} = ෍ 𝑀 = ෍

𝑜𝑢𝑡Ԧ𝑟 × ሶ𝑚𝑉 − ෍

𝑖𝑛Ԧ𝑟 × ሶ𝑚𝑉

Steady state+uniform flow+same axis

𝑇_{𝑠𝑦𝑠𝑡𝑒𝑚} = ෍ 𝑀 = ෍

𝑜𝑢𝑡𝑟 ሶ𝑚𝑉 − ෍

𝑖𝑛𝑟 ሶ𝑚𝑉

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The Angular-Momentum Principle

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The Angular-Momentum Principle

Mass conservation

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