The Conservation of Energy II-1

THE CONSERVATION OF ENERGY FOR A CONTROL VOLUME

Apply the first law of thermodynamics over a control volume bounded by a control surface. This

can be applied either in terms of rates or in terms of amounts of energies over a fixed period of

time.

In terms of rates: The rate at which thermal and mechanical energy enters a control volume,

plus the rate at which thermal energy is generated, minus the rate at which thermal and

mechanical energy leaves the control volume must equal the rate of increase of energy stored

within the control volume.

stst

outgin Edt

dEEEE &&&& ≡=−+

outin EandE && include thermal (internal energy) and mechanical energy (potential and kinetic)

and are associated with surface phenomena and are proportional to the area.

gE& is associated with the rate of conversion from some other energy (chemical,

electrical RIEg2=& , electromagnetic or nuclear) to thermal energy. It is a

volumetric phenomenon, VolumeEg ∝&

stE& is also a volumetric phenomenon, VolumeEst ∝& . It is associated with increase or

decrease of the energy of the matter occupying the control volume that includes

the internal and mechanical energy (kinetic and potential).

If 0=stE& then 0=−+ outgin EEE &&& (steady-state).

The Conservation of Energy II-2

In terms of energies over a time interval t∆ : The amount of thermal and mechanical energy

that enters a control volume, plus the amount of thermal energy that is generated, minus the

amounts of thermal and mechanical energy that leaves the control volume must equal the

increase in the amount of energy stored in the control volume.

stoutgin EEEE ∆=−+

The above equations can be used to develop more specific forms of the energy conservation

requirement. For a closed system of fixed mass, where there is no energy conversion and

changes in kinetic and potential energy are negligible, the previous equation can be written.

UWQ ∆=−

or in terms of rates

dt

dUWq =− &

The other form of energy equation you are familiar (from CHBE 251 – Fluid mechanics) is

applicable for an open system under steady-state.

0=−+⎟⎟⎠

⎞⎜⎜⎝

⎛++⎟⎟

⎠

⎞⎜⎜⎝

⎛++ Wq gz+

2Vpum - gz+

2Vpum

o

2

i

2&&& υυ

Note that the internal energy term and the flow work can be replaced by the enthalpy term

υpui += .

Many other forms of the above equation have been derived and discussed in CHBE251,

(Transport phenomena I – Fluid mechanics).

The Conservation of Energy II-3

The Conservation of Energy II-4

The Conservation of Energy II-5

The Conservation of Energy II-6

The Conservation of Energy II-7

THE SURFACE ENERGY BALANCE

Frequently it is very useful to apply the conservation of energy at the surface of a medium. In

this special case the control surface includes no mass or volume and therefore no generation or

storage terms exist.

The conservation of energy takes the form

0=− outin EE &&

Even though thermal energy generation may be occurring in the medium, the process would not

affect the energy balance at the control volume. Moreover, this equation holds for steady-state

and unsteady-state (transient) conditions

From Figure above

0""" =−− radconvcond qqq

Where

dxdTkqcond −="

)( 2"

∞−= TThqconv

)( 442

"surrad TTq −= εσ

Thus

)()( 4422 surTTTTh

dxdTk −+−=− ∞ εσ

The Conservation of Energy II-8

The Conservation of Energy II-9

The Conservation of Energy II-10

Analysis of HT problems

• State briefly what is known (in your own language)

• State briefly what must be found

• Draw a schematic of the system (control volumes/faces)

• List simplifying assumptions

• Compile physical properties

• Analyze using conservation laws and rate equations

• Discuss the results

The Conservation of Energy II-11

EXAMPLE

A closed container filled with hot coffee is in a room whose air and walls are at a fixed

temperature. Identify all heat transfer processes that contribute to cooling of the coffee.

Comment on the features that would contribute to a superior container design.

The Conservation of Energy II-12

PROBLEM: Radioactive wastes are packed in a long, thin-walled cylindrical container. The

wastes generate thermal energy nonuniformly according to the relation ( )[ ]20/1 rrqq o −= && , where

q& is the local rate of energy generation per unit volume, 0q& is a constant, and or is the radius of

the container. Steady-state conditions are maintained by submerging the container in a liquid that

is at ∞T and provides a uniform convection coefficient h. Obtain an expression for the total rate at

which energy is generated in a unit length of the container. Use this result to obtain an expression

for the temperature Ts of the container wall.

The Conservation of Energy II-13

SUMMARY