Hydrostatic Pressure in Liquids

3specific weight Ngm

γ ρ = = =

82 45.67 10 W

m Kσ − = ⋅

Power= Force * Velocity = VI

( )2 1 2 1p p z zγ− = − −

F pdA= ∫

updownp p zγ= + ∆

Any two points at the same elevation in a continuous mass of same static fluid will be at the same pressure. Accelerating liquids

1tan angle between horiz and surfacex

z

ag a

θ −= =+

Buoyancy

Volume Displaced BF γ= ⋅

1

immersed

BB B Bimmersed V

x x x xdVV′ ′= − = ∫

tanBB O

submurged

x IMB MG GB

Vθ′= = = +

stable if 0MG >

2 2

original waterlineO

waterline

I x dA x dA= ≈∫ ∫

Reynolds transport theorem

velocity, mass, outward normalrelv M n= = =r r

( )CMrel

CV CS

dB dbdV b v n dA

dt dtρ ρ= + ⋅∫ ∫

r r Bb M=

Conservation of mass

( )relCV CS

ddV v n dA

dtρ ρ= − ⋅∫ ∫

r r

CVin out

in out

dMm m

dt= −∑ ∑& &

( )cv cs

system

dmdA

t dtρ

ρ∂ + = ∂ ∫ ∫ V ng

Linear Momentum

unit vector, normal vector, outward positiveun n= =r r

( ) ( )

( )

uCV CS CS

relCV CS

dvdV Pn dA dA

dt

gdV v v n dA

ρ τ

ρ ρ

= − + +

− ⋅

∫ ∫ ∫

∫ ∫

r r r

r r r r

( ) ( )

( ) ( )

rel uCV CS CS

rel relin outin outCV

dv dV Pn dA dA

dt

gdV mv mv

ρ τ

ρ

= − + +

+ −

∫ ∫ ∫

∑ ∑∫

r r r

r r r& &

( )rel relCV CS

dF v dV v v n dA

dtρ ρ= + ⋅∑ ∫ ∫

r r r r

Bernoulli Equation Important to approximate the unsteady term.

absolute gauge atmP P P= +

( ) ( )2 2 2 2

2 1 2 11 1

1 10

2v

ds dp v v g z zt ρ

∂+ + − + − =

∂∫ ∫r

r r

( ) ( )2 2

2 12 12 1 0

2

v vp pg z z

ρ

−−+ + − =

r r

Angular Momentum

( ) ( ) ( )

( ) ( )

rel surfaceCV CV

shaft rel relCS

d r v dV r F r g dVdt

T r v v n dA

ρ ρ

ρ

× = × + × +

− × ⋅

∑∫ ∫

∫

rr r r r r

r r r r r

( ) ( ) ( )

( ) ( )

rel surface shaftCV CV

in rel out relin outin out

dr v dV r F r g dV T

dt

m r v m r v

ρ ρ

× = × + × + +

× − ×

∑∫ ∫

∑ ∑

r rr r r r r

r r r r& &

( )( ) ( ) ( )o rel rel relcv cs

dv dV v v dA

dtρ ρ= × + ×∑ ∫ ∫M r r n

r r rg

Pipe Flows

eRvd vdρµ ν

= =r r

pipe diameterd =

µν

ρ=

2300tran =

2 21 21 1 1 2 2 2

1 12 2 f

p pv gz v gz ghα α

ρ ρ+ + = + + +

r r

2

2fL v

h fd g

=r

2

8 wfvτ

ρ= r

2

2fL v p

h f zd g gρ

∆= = ∆ +

r

2

, 4

64 1282f lam

L v LQh

vd d g gdµ µ

ρ πρ= =

rr

1.111 6.91.8log

Re 3.7d

df

ε ≈ − +

( )3 2

2

ReRe

2f d

d

gd h ffcn

Lξ

ν= = =

1.775Re 8 log

3.7d

dεξ

ξ

= − +

Q udA= ∫ First Law

( )2 1 0iso

E E− = ( )2 1 1 2 1 2E E Q W− −− = −

( )uncoupled sys (coupled) kin ela grav ther coupledE E E E U U= + + + +

W F drδ− = ⋅ 2

1

1 2

r

r

W F dr−− = ⋅∫

1 2 1 2

dUQ W

dt− −− =& &

Constitutive Relations

212kinE mv=

dvF ma m

dt= =

2

1

1 2

v

v

W mvdv−− = ∫

212elaE kx= F kx=

graE mgh= F mg=

1 1theU CT mcT= = 22 1

1

lnT

S S mcT

− =

PV mRT= 1revW PdVδ = 2

1

1 2 1

V

V

W PdV− = ∫

gas vU mc T= p vc c R= + p

v

c

cγ = constantPV γ =

vMa

RTγ=

r

2 22 1

1 1

ln lnp

T PS S mc mR

T P

− = −

2 22 1

1 1

ln lnv

T VS S mc mR

T V

− = +

2 22 1

1 1

ln lnv p

P VS S mc mc

P V

− = +

Reversible adiabatic

11

2 21 2 1 1

1 1

1 1v vP V

W m c T mc TP V

γγ

γ−

−

−

− = − = −

Second Law

( )2 1 0 genisoS S S− ≥ =

2

1

A

A

ABtrans

A

QS

Tδ

= ∫

( )2

2 11

A

A

genA

QS S S

Tδ

− = +∫

Flow Field

2 2 2

2 2 2x

p u u u dug

x x y z dtρ µ ρ

∂ ∂ ∂ ∂− + + + = ∂ ∂ ∂ ∂

2 2 2

2 2 2y

p v v v dvg

y x y z dtρ µ ρ

∂ ∂ ∂ ∂− + + + = ∂ ∂ ∂ ∂

2 2 2

2 2 2z

p w w w dwg

z x y z dtρ µ ρ

∂ ∂ ∂ ∂− + + + = ∂ ∂ ∂ ∂

xyu vy x

τ µ ∂ ∂

= + ∂ ∂ xz

w ux z

τ µ∂ ∂ = + ∂ ∂

zyv wz y

τ µ ∂ ∂

= + ∂ ∂

( )0

x

shear wF b x dxτ= ∫

Heat Transfer Conduction

c

dTQ kA

dx = −

& W

kmK

= ( )1 2

ckAQ T T

L= −&

Convection

( )c s sQ h A T T∞= −&

Radiation

( )1 2r sQ h A T T= −& 314r mh Tε σ= 1 2

2m

T TT

+ =

Thermal Resistance

1 2

total

T TQ

R −

=

& convection in parallel with radiation

cond

LR

kA=

1conv

c

Rh A

= 1

radr

Rh A

=

,

ln

2

o

icondradial

RR

RkLπ

=

Heat Diffusion Equation alpha = thermal diffusivity Cartesian

genq =& rate of energy dissipated per unit mass by all energy storage

methods other than thermal. q =& rate of energy generated ie

genq

qρ

=&

&

2 2 2

2 2 2

T k T T T qt c x y z cρ ρ

∂ ∂ ∂ ∂= + + + ∂ ∂ ∂ ∂

&

kc

αρ

=

Cylindrical

2

1 1p

T T T Tc kr k k q

t r r r r z zρ

φ φ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂

&

Spherical

22 2 2 2

1 1 1sin

sin sinpT T T T

c kr k k qt r r r r r

ρ θθ φ φ θ θ θ

∂ ∂ ∂ ∂ ∂ ∂ ∂ = + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ &

Entropy Generation Equation 22 2

2gen

gen

qk T T Ts

T x y z Tρ

∂ ∂ ∂ = + + + ∂ ∂ ∂

&&

gen genS s dxdydzρ=& & s=per unit mass

Fin Temperature Distributions

T Tθ ∞≡ − ( )0b bT Tθ θ ∞= = − 2

c

hPm

kA≡

c bM hPkA θ≡

Convection ( )x L

dh L k

dxθ

θ=

= −

( )( )

cosh ( ) sinh ( )

cosh sinhb

hm L x m L xmkhmL mLmk

θθ

− + −=

+

( )( )

sinh cosh

cosh sinhf

hmL mLmkq MhmL mLmk

+=

+

Adiabatic

0x L

ddxθ

=

= cosh ( )

coshb

m L xmL

θθ

−= tanhfq M mL=

Prescribed Temperature

( ) LLθ θ=

sinh sinh ( )

sinh

L

b

b

mx m L x

mL

θθθ

θ

+ − =

cosh

sinh

L

bf

mLq M

mL

θθ−

=

Infinite fin

( ) 0Lθ = mx

b

eθθ

−= fq M=

Fin Efficiency

max

f ff

f b

q qq hA

ηθ

= =

Total array efficiency

exposed surfacetotal finA NA A= + max c total bQ h A θ=&

( )1 1finsurface fin

total

NA

Aη η= − − maxtotal surfaceQ Qη=& &

Biot Number

cs

VL A=

internal conduction resistance1external convection resistance

solid c c c

solidc s

Lk A h L

Bik

h A= = ≈

1 2 1 2

dUQ W

dt− −− =& &

1Bi = Lumped thermal capacitance model Reversible in terms of the solid. Principal resistance to heat transfer lies within the fluid. Temperature of solid can be modeled as uniform at all times even though it is changing in time.

kc

αρ

= ( )1th th

c s c s

cVcV R C

h A h Aρ

τ ρ

= = =

c sh At

BiFocV

i

e eρθθ

−− ⋅= = 2

c c

t tFo

L tα

= = tc is time for

disturbance to diffuse characteristic length

2 2c s c c c solid c c

c solid c solid c

hAt ht hL k hLt tBi Fo

cV cL k c L k Lα

ρ ρ ρ

= = = = ⋅

( )c s

dTcV h A T T

dtρ ∞= − − ( ) 0gen solid

S =

( ) ( ) ( )2

ln 12

solidgen solidfluid

cV T T TS cV

T T T

ρρ

∞ ∞ ∞

∆ ∆ ∆= = − +

( )0 0

( ) lnfinal finalt t

fc strans solid

i

Th A T TQS dt dt cV

T T Tρ∞

=

− = =

∫ ∫

&

Bi → ∞ Fluid behaves as heat reservoir. Principal resistance to heat transfer lies within the solid. Temperature of fluid can be modeled as uniform at all times even though it is changing in time.

( )2

212

0

2 1 1cos

1 22

nn Fo

nb

xe n

Ln

πθπ

θ π

∞ − +

=

− = + +

∑

2

22 ( ) .2

Fosolid i

c

k T TQe Fo

A L

π − ∞ −= ≥

&

( ) .05solid i

c

k T TQFo

A tπα∞−

= ≤&

Semi Infinite Model

2

2

T Tt x

α∂ ∂

=∂ ∂

with boundary conditions

i

s i

T TT T

θ−

=−

4x

tη

α=

2

0

2 1uerf e du erfc erf

η

η η ηπ

−= = −∫

Case I Constant Surface temperature

i

s i

T Terfc

T Tθ η

−= =

−

( )solid s i

c

k T TQA tπα

−=

&

Case II Constant Heat Flux at Surface

2

444

xt

iQ t x

T T e xerfcAk t

ααπ α

− − = −

&

Case III Convective Heat Transfer at the Surface

2c ch x h

tk ki c

s i

T T herfc e erfc t

T T k

α

η η α

+ − = − + −

Case IV Surface Energy Pulse

2

4x

ti

ET T e

c tα

ρ πα

−− =

Case V Periodic Variation of Surface Temperature

2

2xi

s i

T Te sin t x

T Tω α ω

ωα

− −= −

−

Two Semi -Infinite Solids in Simple Thermal Communication

( ) ( )( ) ( )

, ,A i B iA Bs

A B

T k c T k cT

k c k c

ρ ρ

ρ ρ

+=

+

Reversible Cycles Carnot Cycle

0H Ltransfer

H L

Q QQS

T T Tδ

δ = = + =∫ ∫Ñ Ñ

( ) 1H LH L H

H H

Q TW T T Q

T Tδ

= − = −

∫Ñ

H

W

Q

δη = ∫Ñ

11

LH

H Lrev

H H

TQT T

Q Tη

−

= = −

1

1

L

H

L

QCOP

QWQ

δ= =

− − −∫Ñ

1

1rev

H

L

COPTT

=−

H HHP

H L

Q QCOP

Q QWδ= =

+∫Ñ ( ) 1

1HP rev

L

H

COPTT

=−

The energy transferred in the form of a heat transfer from a higher temperature source has a higher “quality” (greater value for energy conversion purposes) than the energy transferred as a heat transfer from a lower temperature source because it carries less entropy.

0H Ltransfer gen gen

H L

Q QS S S

T Tδ δ δ+ = + + =∫ ∫ ∫Ñ Ñ Ñ

( ) ( ) L genrev irrevW W T Sδ δ δ− =∫ ∫ ∫Ñ Ñ Ñ

Lirrev rev gen

H

TS

Qη η δ= − ∫Ñ

1

1irrev

H Hgen

L L

COPT T

ST Q

δ=

− + ∫Ñ

Temperature Distributions Plane Wall (x=0 at middle of wall, x= +L or –L at ends of wall, -L=T1

2 2,2 ,1 ,1 ,2

2( ) 12 2 2

s s s sT T T TqL x xT x

k L L− +

= − + +

&

( ) ( ),2 ,12 s s

kq x qx T T

L′′ = − −&

( ) ( ),2 ,12 s s x

kq x qx T T A

L = − −

&

Cylindrical Wall

( ) ( )( )

2 2 2222 2 1

,2 ,2 ,12 22 2 2 1

ln( ) 1 1

4 4 lns s s

r rqr qr rrT r T T T

k r k r r r

= + − − − + −

& &

( )

( )

2 22 1

,2 ,122

2 1

14

( )2 ln

s sqr rk T T

k rqrq rr r r

− + −

′′ = −

&&

( )2 1ln2cond

r rR

Lkπ=

Spherical Wall

( )2 2 22

2 2 1 2,2 ,2 ,12 2

2 2 1 2

1 1( ) 1 6 1

6 4 1 1s s s

qr qr r r rrT r T T T

k r k r r r

−= + − − − + − −

& &

( )

[ ]

2 22 1

,2 ,122

21 2

16

( )3 1 1

s sqr rk T T

k rqrq rr r r

− + −

′′ = −−

&&

1 2

1 1 14condR

k r rπ

= −

Summary of Governing Relations for a Control Volume

t h e r m a l e n e r g yu = 1v= = s p e c i f i c v o l u m eρ

J

h u Pvkg

= +

H U PV mh= + =

Reynolds Transport Theorem

( )CM

CV CS

dB dbdV b n dA

dt dtρ ρ ϑ= + ⋅∫ ∫

r r Bb M=

Conservation of Mass (Continuity Equation)

( )CV CS

ddV n dA

dtρ ρ ϑ= − ⋅∫ ∫

r r CV

in outin out

dMm m

dt= −∑ ∑& &

First Law of Thermodynamics

( )2 2

2 2shaft rCV CS

d v vh gz dV Q W h gz v n dAdt

ρ ρ + + = − − + + ⋅ ∫ ∫

r r& &

2 2

2 2CV

shaft in outin outin out

dE v vQ W m h gz m h gz

dt

= − + + + − + +

∑ ∑& & & &

Second Law of Thermodynamics

( ) geniCV CSi

d QsdV s n dA S

dt Tρ ρ ϑ

= − ⋅ +

∑∫ ∫

& r r &

( ) ( )CVgenin out

i in outi

dS Qms ms S

dt T

= + − +

∑ ∑ ∑&

&& &

Linear Momentum Equation

( ) ( ) ( )rCV CS CS CV CS

d dV Pn dA dA gdV n dAdt

ρϑ ρ τ ρ ρϑ ϑ= − + + + ⋅∫ ∫ ∫ ∫ ∫r r rr r r r

( ) ( ) ( ) ( )in out

in outCV CS CS CV

d dV Pn dA dA gdV m mdt

ρϑ ρ τ ρ ϑ ϑ= − + + + −∑ ∑∫ ∫ ∫ ∫r r rr r r & &

Angular Momentum Equation

( ) ( ) ( ) ( ) ( )surface shaft rCV CV CS

d r dV r F r g dV T r n dAdt

ϑ ρ ρ ϑ ρ ϑ

× = × + × + − × ⋅

∑∫ ∫ ∫r r rr rr r r r r r

( ) ( ) ( ) ( ) ( )surface shaft in outin outin outCV CV

d r dV r F r g dV T m r m rdt

ϑ ρ ρ ϑ ϑ

× = × + × + + × − ×

∑ ∑ ∑∫ ∫r r rr rr r r r r r& &

general coordinate information dA rdrdθ= quasi static= passing through a series of equilibrium states.

Accelerationm

s2 Area m2

Densitykg

m3 Energy J

J 1 kg m2 s 2= Force N N 1 kg m s 2= HeatTransferRate W W 1 kg m2 s 3= Heatflux

W

m2

HeatGenRateW

m3 HeatTransferCoeff

W

m2 K.

KViscositym2

s LatentHeat

J

kg

J

kg1 m2 s 2=

Length m

Mass kg MassDensitykg

m3

MassFlowRatekg

s MassTransferCoef

1

m

Power W W 1 kg m2 s 3= PressureStress

N

m2 SpecificHeat

J

kg K.

Temperature K TempDiff K ThermalCond

W

m K. W

m K.1 kg m s 3 K 1=

ThermalResistK

W K

W1 kg 1 m 2 s3 K=

DViscosityN s.

m2 Volume m3

VolumeFlowRatem3

s

Definitions and such When to use Bernoulli.

1. Steady flow. (otherwise, use unsteady version and estimate unsteady term.

2. Incompressible flow- Mach number less than .3. 3. Frictionless flow

4. Flow along a single streamline 5. No shaft work-no pumps or turbines. 6. No heat transfer, ie adiabatic.

Heat Transfer The energy transfer interaction that occurs between pure thermal systems Temperature The property of a system which indicates the potential for heat transfer with other systems. Two systems are unequal in temperature when they experience heat transfer. For a heat transfer in the absence of work transfer, the heat transfer is positive for the low temperature system which increases in energy and negative for the high temperature systems which decreases in energy A quasi -static process is a model for a dynamic process in which the state of the system is changing at a rate which is slow compared to the rate at which the system approaches the equilibrium state by means of energy and entropy transfer processes internal to the system boundary. Internally, the system appears to be in equilibrium at all times throughout the process even though its state is changing with time. The equilibrium properties thus provide a complete description of the state of the system at all times throughout the process. A reversible process generates no entropy, is a sequence of equilibrium states, and proceeds in the reverse direction just as readily as in the forward direction and takes an infinitely long time to be carried out. Head loss is the change in the sum of the pressure and gravity head, the change in height of the hydraulic grade line or height change of the energy grade line System: A system is defined as an quantity of matter or region of space to which attention is directed for purpose of analysis. Boundary: The quantity of matter or region of space which forms the system is delineated by a boundary, a surface either real or imaginary. State and Property: State is used to signify the condition of a system at a specific instant. The state of a system is characterized by a collection of observable, macroscopic quantities, called properties. A property is one of those observable macroscopic quantities which are definable at a particular instant without reference to the system’s history.