AME$60634$$ Int.$HeatTrans.$ Work Examplessst/teaching/AME60634/lectures/AME...1545...

AME 60634 Int. Heat Trans.

D. B. Go 1

Work Examples

F

CM

Δx [1] Sliding Block

work done to the control mass so it is energy gained

€

W = F ⋅ Δx Nm[ ] = J[ ]

[2] Shear Work on a Fluid Belt

τ CM

Liquid Bath

W vx

€

˙ W = τ ⋅ vx ⋅ A Nm2

ms

m2$

% & '

( ) =

Js$

% & '

( ) = W[ ]


shear stress × speed × area


D. B. Go 2

Work Examples

p1

p0

CM

W

[3] Boundary Displacement

Δz

€

W = F ⋅ ΔzW = p1 − p0( )A ⋅ Δz

work done by the control mass so it is energy lost

boundary work

W = pdVVi

Vf

∫ J[ ]

Gas Expansion

Strain (Compression/Expansion)

CM1

F

Δz

€

W = F ⋅ ΔzW =σA ⋅ Δz


boundary work

W = σ dVVi

Vf

∫ = σ Adzzi

z f

∫ J[ ](constant area)


D. B. Go 3

Work Examples [4] Shaft/Propeller

W = T ⋅ θ W[ ]

[5] Electrical Work (Heat Generation)

W

CM torque × angular speed


CM

+ - W

W = i ⋅V =V 2

R= i2R W[ ]

Joule (or resistive or Ohmic) heating


V

R


D. B. Go 4

Work Examples [6] Surface Tension

€

W = γ ⋅ ΔANm

m2%

& ' (

) * = J[ ]

surface tension × area change


Soap bubble

air

CM

straw

CM

movable wire

Soap film inside a wire

ΔA


D. B. Go 5

Work Examples [7] Spring Compression

F = kxdW = Fdx = kxdx

W = kx dx = 12k x f

2 − xi2( )

xi

x f

∫ J[ ]

F

Δx


D. B. Go 6

Enthalpy We can literally define a new specific property enthalpy as the summation of the internal energy and the pressure × volume (flow work)

Porter, 1922

h = u+ pv→H =U + pV

Thus for open systems, the first law is frequently written as

dECV

dt= Q− Wnet + min h+ 1

2vx2 + gz

"

#$

%

&'inin

∑ − mout h+12vx2 + gz

"

#$

%

&'outout

∑


D. B. Go 7

Property, State, and Process •  Property is a macroscopic characteristic of the system •  State is the condition of the system as described by its properties. •  Process changes the state of the system by changing the values of

its properties –  if a state’s properties are not changing then it is at steady state –  a system may undergo a series of processes such that its final and

initial state are the same (identical properties) – thermodynamic cycle

•  Phase refers to whether the matter in the system is vapor, liquid, or solid –  a single type of matter can co-exist in two phases (water and steam) –  two types of matter can co-exist in a single phase (a water/solvent

mixture) •  Equilibrium state occurs when the system is in complete

mechanical, thermal, phase, and chemical equilibrium è no changes in observable properties


D. B. Go 8

Properties •  extensive properties (dependent on size of system)

–  U internal energy [kJ] H enthalpy (total energy) [kJ] –  V volume [m3] m mass [kg] –  S entropy [kJ/K]

•  intensive properties (independent of size of system) –  ρ density [kg/m3] –  T temperature [K] –  p pressure [Pa] –  x quality [-]

•  specific properties: the values of extensive properties per unit of mass of the system [kg-1] or per unit mole of the system [kmol-1] (inherently intensive properties) –  u specific internal energy [kJ/kg] h specific enthalpy [kJ/kg] –  v specific volume [m3/kg] –  s specific entropy [kJ/(kg-K)]

h = u+ pv


D. B. Go 9

Pure Substances, Compressible Systems

seek a relationship between pressure, specific volume, and temperature •  from experiment it is known that temperature and specific volume are

independent •  can establish pressure as a function of the others p = f (v,T )

p-v-T surface water

p-v-T Relationship

single phase: all three properties are independent (state fixed by any two)

two-phase: properties are dependent on each other (state fixed by specific volume and one other)

•  occurs during phase changes

saturation state: state at which phases begins/ends


D. B. Go 10

Pure Substances, Compressible Systems p-v-T Surface Projections

phase diagram p-v diagram

•  two-phase regions are lines •  triple line is a triple point •  easily visualize saturation

pressure & temperature

•  constant temperature lines (isotherms)


D. B. Go 11

Pure Substances, Compressible Systems p-v-T Surface Projections T-v diagram

•  constant pressure lines (isobars) •  quality x denotes the ratio of vapor to total mass in two-phase mixture

x =mvapor

mvapor +mliquid

v = 1− x( )vf + xvg

two-phase properties from saturation properties


D. B. Go 12

Phase Changes •  vaporization/condensation – change from liquid to gas and vice versa

•  only occurs below critical point •  above critical point, the distinction between the two states is not clear

•  melting/freezing – change from solid to liquid and vice versa •  only occurs above triple point •  below triple point, the liquid state is not possible and solids change directly

to gas (sublimation)


D. B. Go 13

Evaluating Liquid Properties

v(T,p) ≈ vf(T) u(T,p) ≈ uf (T) h(T,p) ≈ uf (T)+pvf(T)

For liquids, specific volume and specific internal energy are approximately only functions of temperature

(saturated liquid)

When the specific volume v varies little with temperature, the substance can be considered incompressible

cv =∂u∂T"

#$

%

&'v

=dudT

it follows h = u T( )+ pv→ cp =∂h∂T#

$%

&

'(p

=dudT

thus cp = cv incompressible liquids

Changes in u and h can be found by direct integration of specific heats


D. B. Go 14

Compressibility Factor Compressibility Factor Z = pv

RT8.314 kJ/kmol·K 1.986 Btu/lbmol·oR 1545 ft·lbf/lbmol·oR

universal gas constant =R R = R

M (molecular weight)

At states where the pressure p is small relative to the critical pressure pc (where pR is small), the compressibility factor Z is approximately 1.

Virial equations of state: Z =1+ B T( ) p+ C T( ) p2 + D T( ) p3 +...


D. B. Go 15

Evaluating Gas Properties

At states where the pressure p is small relative to the critical pressure pc (where pR is small), the compressibility factor Z is approximately 1.

Z =1→ pv = RT ideal gas

u(T,p) ≈ u(T) h(T,p) ≈ u(T)+pv = u(T)+RT

For ideal gas, specific internal energy and enthalpy are approximately only functions of temperature

≈ h(T)

cv =dudT

Specific heat cp =dhdT

Changes in u and h can be found by direct integration of specific heats

and


D. B. Go 16

Heat Transfer •  Heat Transfer is the transport of thermal energy due to a

temperature difference across a medium(s) –  mediums: gas, liquid, solid, liquid-gas, solid-gas, solid-liquid, solid-solid,

etc. –  Thermal Energy is simply the kinetic energy (i.e. motion) of atoms and

molecules in the medium(s)

•  Atoms/molecules in matter occupy different states –  translation, rotation, vibration, electronic –  the statistics of these individual molecular-level activities will give us

the thermal energy which is approximated by temperature

•  Heat Transfer, Thermal Energy, and Temperature are DIFFERENT. DO NOT confuse them.

•  Heat generation (electrical, chemical, nuclear, etc.) are not forms of heat transfer Q but forms of work W –  Q is the transfer of heat across the boundary of the system due to a

temperature difference


D. B. Go 17

Definitions

Thermal Energy

Temperature

Heat Transfer

Energy associated with molecular behavior of matter

U [J] – extensive property u [J/kg] – intensive property

Means of indirectly assessing the amount of thermal energy stored in matter

Quantity Meaning Symbol/Units

T [K] or [°C]

Thermal energy transport due to a temperature gradient (difference)

various

Heat

Heat Rate/Heat Flow

Heat Flux

Thermal energy transferred over a time interval (Δt > 0)

Thermal energy transferred per unit time

Thermal energy transferred per unit time per unit surface area

Heat Transfer

€

q, ˙ q , ˙ Q [W]

€

" " q [W m2]

€

Q [J]


D. B. Go 18

Modes of Heat Transfer

•  Conduction & convection require a temperature difference across a medium (the interactions of atoms/molecules)

•  Radiation transport can occur across a vacuum

AME$60634$$ Int.$HeatTrans.$ Work Examplessst/teaching/AME60634/lectures/AME...1545...

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