Quiz 2 – 2013.11.27

Questions

1. What is the Reynolds number?2. Differentiate the flow patterns observed in laminar

flow from those in turbulent flow.3. How does temperature affect the dynamic viscosity

of a fluid?

TIME IS UP!!!

Overall Balances

Outline

1.Mass Balance

2.Energy Balance

3.Momentum Balance

Mass Balance

For an overall mass balance, no mass is being generated. Why?

rate of mass output rate of mass input from control volume to control volume

rate of accumulation 0of mass within control volume

Mass Balance

Imagine the control volume as having infinitesimal surfaces dA. We need to find the net outflow of mass across the control surface.

Mass Balance

For every dA element, a streamline of velocity vector v passes through it.

Mass Balance

For every dA element, a unit normal vector n exists.

Mass Balance

The component of velocity vector v in the direction of the unit normal vector n is given by: cosv n vn

Mass Balance

The rate of mass efflux through dA:

vn A v n A ( )d cos d

Mass Balance

What do we get when we integrate over the entire control surface?

vn A d

Mass Balance

POSITIVE: net outflow of massNEGATIVE: net inflow of massZERO: ?

vn A d

Mass Balance

Rate of mass outflow across control surface (and control volume):

Rate of mass accumulation in control volume:

vn A d

dVt

Mass Balance

rate of mass output rate of mass input from control volume to control volume

rate of accumulation 0of mass within control volume

vn A Vt

d d 0

Overall Mass Balance

vn A Vt

d d 0

Mmt

d 0d

A well-stirred storage vessel contains 10000 kg of dilute methanol solution (xMetOH = 0.05). A constant flow of 500 kg/min of pure water is suddenly introduced into the tank and a constant rate of withdrawal of 500 kg/min of solution is started. These two flows are continued and remain constant. Assuming that the densities of the solutions are the same and that the total contents of the tank remain constant at 10,000 kg of solution, calculate the time for the alcohol content to drop to 1.0 wt.%.

Overall Mass Balance

Outline

1.Mass Balance

2.Energy Balance

3.Momentum Balance

• Possessed/Carried by fluid– Internal Energy– Potential Energy– Kinetic Energy– PV-work

• Transferred between system and surroundings– Heat– Shaft work

Forms of Energy

• Intrinsic property of the fluid• Molecules in random motion

Internal Energy (U)

• Position of the fluid with respect to an arbitrary reference plane

cggz

Potential Energy (mgz)

• Due to fluid motion• Correction factor, a– To account for velocity distribution– Ranges from 0.5 (laminar) to 1.0 (turbulent)

Kinetic Energy (mv2/2α)

• Work done by surroundings to push the fluid into the system

P S

d

PVSVPSFdWPV

PV Work (PV)

• Net heat passing through the boundary of the system– Positive if heat is transferred to the system from

the surroundings – Negative if system to the surroundings

• Excludes heat generated by friction

Heat (Q)

• Net work done on the system by the surroundings

• Convention (IUPAC)– Positive if work done on the system– Negative if work done by the system

Shaft Work (Ws)

Energy balance from point 1 to point 2:

Datum/reference plane

U1, v1, P1, V1, S1

U2, v2, P2, V2, S2

z1

z2

Q

Ws

Total Energy Balance

21

1 1 1 1

22

2 2 2 2

2

2( )

s

vm U gz PV

Q Wvm U gz PV

dmUdt

a

a

Energy balance from point 1 to point 2:

Total Energy Balance

2 21 2

1 1 1 1 2 2 2 2( )

2 2sv v dmUm U gz PV Q W m U gz PV

dta a

2( )2 s

dmU vm U gz PV Q Wdt a

2( )2 s

dmU vm H gz Q Wdt a

Water at 93.3°C is being pumped from a large storage tank at 1 atm abs at a rate of 0.189 m3/min by a pump. The motor that drives the pump supplies energy at the rate of 1.49 kW. The water is pumped through a heat exchanger, where it gives up 704 kW of heat and is then delivered to a large open storage tank 15.24 m above the first tank. What is the final temperature of the water to the second tank?

Total Energy Balance

A modification of the total energy balance- shaft work- kinetic energy- potential energy- flow work (PV)

Does not include heat and internal energy. - Why?

Energy converted to heat is lost work- loss of mechanical energy by friction

Mechanical Energy Balance

No shear stress; zero viscosity

For isothermal flow and Q=WS=0,

Bernoulli Equation

2

22

221

21

11 22gzvVPgzvVP

aa

2

22

2

21

21

1

1

22z

gv

gPz

gv

gP

aa

Ideal Fluids

Restrictions:

1. Valid only for incompressible fluids 2. No devices that add/remove energy should

be between points 1 and 23. No heat transfer occurring in the system4. No loss of energy due to friction

2

22

2

21

21

1

1

22gz

gv

gPz

gv

gP

aa

Bernoulli Equation

• Friction losses: SF (energy dissipation)• Total heat absorbed by the fluid• Total work done by fluid,

-W = -WS + SF– Additional work must be done by the fluid to

overcome fluid friction

Real Fluids

Q Q F

• Note: energy per mass units• kJ/kg or ft-lbf/lbm

• For incompressible flow:

2

2 Sv Pg z F Wa

S

2

( ) 2SWv Q FU PV gz

m ma S

Real Fluids

2

2sWv QU gz PV

m ma

Q Q FQ Q F