1

Lec 3: Conservation of mass continued, state postulate,

zeroth law, temperature

2

• For next time:– Read: § 10-1 to 10-2, 10-4 to 10-5, and 2-5 to 2-8.

• Outline:– Conservation of mass example problems– Equilibrium and states– Zeroth law of thermodynamics

• Important points:– Problem solving methodology– State, path, and process– Temperature scales

3

Review

n

1iim

k

1jjm

dtdmc.v.

C.V. INTO RATEFLOW MASS

C.V. OF OUT RATE

FLOW MASS

C.V. THE IN MASS OF

CHANGE OF RATE

where

vAρm VVA

and cvcv ρVm

4

Steady flowFlows are steady if any derivative with respect to time equals zero:

0dt

dmCV

outletsall

einletsall

i mm

The term steady state may also be used, which simply means that any property (used to define a state) does not vary with time, although they may vary with position.

5

Conservation of mass

21 A)(A)( VV

21 )A()A(VV

OR

If we limit ourselves to steady-state devices with one entrance and one exit,

6

Look at some simplifying cases:

Incompressible pipe flow:

21 21 AA 21 VV

General Incompressible:

21 2211 AVAV

7

One more simplifying case:

Ideal gas incompressible:

RTP

2

222

1

111

TAVP

TAVP

8

TEAMPLAYTEAMPLAY

Steam enters a turbine with a specific volume of 0.831 ft3/lbm with a velocity of 21.0 ft/s and leaves with a specific volume of 175.8 ft3/lbm. The turbine inlet area is 1 ft2 and the outlet area is 140 ft2.

A) What is the mass flow (lbm/hr)?

B) What is the exit velocity (ft/s)?

9

Review--State

• The state of a system is defined by the values of its properties.

10

TEAMPLAYTEAMPLAY

• How many properties can you name that apply to the gas in a high pressure cylinder of nitrogen?

• How many are independent and how many are dependent?

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Equilibrium

• A system is in equilibrium if its properties are not changing at any given location in the system. This is also known as “thermodynamic equilibrium” or “total equilibrium.”

• We will distinguish four different subtypes

of thermodynamic or total equilibrium.

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Types of thermodynamic equilibrium:

• Thermal equilibrium--temperature does not change with time.

• Mechanical equilibrium--Pressure does not change with time.

• Phase equilibrium--Mass of each phase is unchanging with time.

• Chemical equilibrium--molecular structure does not change with time.

13

Equilibrium

• Equilibrium implies balance--no unbalanced potentials (driving forces) in the system.

14

State Principle or State Postulate

• Text says, “The state of a simple compressible system is completely given by two independent, intensive properties.”

• Properties are independent if one can be constant while the other varies.

• This only applies at equilibrium.

15

Process

• Change in state of a system from one equilibrium state to another.

P

V

1P

V

1P

V

1P

V

1P

V

1

2

16

Path

Series of states through which a system passes.

P

V

1

2

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Properties at end points are independent of the process

P

V

1

2

Path 1

P

V

1

2

Path 1

Path 2P

V

1

2

Path 1

Path 2

P ath 3

18

Constant property processes

• The prefix “iso” is used to indicate a property that remains constant during a process:

– Isothermal is constant temperature– Isobaric is constant pressure– Isochoric or isometric is constant volume

19

Review definitions• Steady state--any property (used to

define a state) does not vary with time, although it may vary with position.

• Compare with definition of equilibrium. A system is in equilibrium if its properties are not changing at any given location in the system.

• So, the question arises: how does something change with time?

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Quasiequilibrium Process

Idealized process in which the departurefrom equilibrium is infinitesimally small.

Gas or liquid system

Boundary

Incremental masses removed during an expansion of the gas or liquid

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Quasiequilibrium Processes

• Engineers are interested in quasiequilibrium processes for two reasons:– They are easy to analyze because many

(relatively) simple mathematical relations apply.

– It will be shown later that devices produce maximum work or require minimum work when they operate on quasiequilibrium processes.

22

Cycle

P

T

A

B

.

.

Path 2

Path 1

Series of processes where the initial and final states are the same.

23

State Principle (more rigorous definition)

• The number of independent, intensive properties needed to characterize the state of a system is n+1 where n is the number of relevant quasiequilibrium work modes.

• This is empirical, and is based on the experimental observation that there is one independent property for each way a system’s energy can be independently varied.

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State Principle continued

• The “1” is for heat transfer (Q).

• The “n” is the number of relevant quasiequilibrium work modes. In this course, we will usually have n = 1.

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Simple system

A simple system is defined as one for which only one quasiequilibrium work mode applies.

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For a simple system,

• We may write: p = p(v,T)

• Or perhaps: v = v(p,T).

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Forms of Energy

• Energy is usually symbolized by E, representing total energy

• e is energy per unit mass

EeM

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Forms of Energy

• Macroscopic forms--possessed with respect to some outside reference frame.– Kinetic energy,

– Potential energy,

2m21KE V 2

21keor V

gz peormgzPE

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Forms of energy

• Microscopic forms are called internal energy (internal to the molecule) and represent the energy a molecule can have as it translates, rotates, and vibrates. There are other contributors--nuclear spin, for example--as well.

• We will not concern ourselves with the details, but will use the symbols U and u.

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Energy

• Now, we have

• and for stationary, closed systems, KE and PE are 0.

• So, for stationary closed systems, E= U

PEKEUE PEKEUE

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Energy

• Sensible energy--the portion of the internal energy associated with all forms of kinetic energy of the molecules.

• Latent energy--refers to internal energy associated with binding forces between molecules. Phase changes, such as vaporizing (boiling) water are latent energy changes.

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Thermal Equilibrium

• Occurs when two bodies are at the same temperature T and no heat transfer can occur.

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Zeroth Law of Thermodynamics

• If two bodies are in thermal equilibrium with a third body, they are in thermal equilibrium with each other.

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Boiling point 100 212 373.15 71.67

ºC ºF K R

Triple point @ 0.006 atm, T = 0.01 ºC

Ice point 0.00 32.00 273.15 491.67

Absolute Zero -273.15 -459.67 0 0

We Need to Work With Temperatures

35

Temperature relationships

• T (ºR) = T (ºF) + 459.67 [use 460]

• T (K) = T (ºC) + 273.15 [use 273]