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Chap. 2 The first law and other basic concepts
The first law is that energy is conserved.
2.4 Energy balance for closed system
No transfer of matter or mass is allowed between a system and its surrounding. Heat and work cross the boundary.
∆U + ∆EK + ∆EP = Q + W
U is internal energy, EK is kinetic energy and EP is potential energy. Q is heat and W is work. Q and W are regarded as positive when they are transferred into the system from the surroundings.
Internal energy (U) = energy associated with all molecular, atomic, and subatomic motions and interactions. It is difficult to determine absolute internal energy. What is important is the difference in internal energy between states.
kinetic energy = ⡩ ⡰ mu
⡰ (^) where m is the mass and u is the speed of the molecules in a system.
gravitational potential energy = mgh where g is the gravitational acceleration and h is the elevation relative to the earth’s surface.
Closed systems often undergo processes during which only the internal energy (U) of the
system changes or ∆EK + ∆EP = 0.
Then, ∆U 㐄 Q ㎗ W in integral form, and dU =dQ + dW in differential form.
(∆U)system + (∆U)surroundings = 0.
Internal energy (J) is an extensive property and molar internal energy (J/mol) is an intensive property.
The mass of the system is constant, but the volume may not be constant. For a gaseous system, the volume is variable unless the system boundary is rigidly fixed.
U is a state function, which is independent of the path by which it reaches a given state. In comparison, Q and W are not state functions and they depend on the path.
(Example 2.1)
Flow work (Wf) = work required to push a material into the system at the inlet or out of the system at the outlet. Wf = Pv where P is the pressure and v is the volumetric flow rate. Shaft work (Ws) = work to rotate the shaft connected to a system. (compressors and pumps
for flow systems) Work due to change in volume (Pressure-Volume work):
Work (W) = (^) ᔖ F • ds ⤩⤰⤑⤰⤕ ⡰ ⤩⤰⤑⤰⤕ ⡩ where F is the force and s is the displacement.
W= (^) ᔖ 䙦PA䙧 • ds ⤩⤰⤑⤰⤕ ⡰ ⤩⤰⤑⤰⤕ ⡩ 㐄 ᔖ^ P • 䙦As䙧
⤩⤰⤑⤰⤕ ⡰ ⤩⤰⤑⤰⤕ ⡩ 㐄 ᔖ^ P • dV
⤩⤰⤑⤰⤕ ⡰ ⤩⤰⤑⤰⤕ ⡩ where P is the pressure. A is the cross sectional area, and V is the volume. By definition in the textbook, dW = -PextdV Pext: external pressure or pressure outside the system
A work by isothermal expansion of gas in multiple steps is illustrated as follows:
The expansion above was carried out in four steps by removing the cubes on the piston one by one. The rectangles with solid boundary lines under the P-V curve represent the amount of work obtained by the expansion and those with the dotted lines represent the amount of work required to return the gas to the initial condition. By increasing the number of the cubes while maintaining the total mass of the cubes constant, the total area of the solid rectangles become closer to that of the dotted rectangles, approaching the area under the P-V curve. With infinite number of cubes or infinite number of steps, the work obtained by expansion is virtually the
at constant pressure, dH = CP dT
∆H 㐄 Q at constant pressure.
∆U = Q at constant volume
The effects of molar volume on internal energy and of pressure on enthalpy are usually small and neglected for simplicity in this chapter. The effects of volume and pressure will be considered later in chapter 6.
Then,
dU 㐄 C⤆dT dH 㐄 C⤀dT
On integration,
∆U 㐄 㔅 C⤆dT
⤄ㄘ
⤄ㄗ
∆H 㐄 㔅 C⤀dT
⤄ㄘ
⤄ㄗ
Example 4.
∆U and ∆H associated with phase changes:
Heat of fusion (solid to liquid) = -heat of solidification (liquid to solid) Heat of vaporization (liquid to vapor) = - heat of condensation (vapor to liquid) Heat of sublimation (solid to vapor)
The Clapeyron equation:
∆H 㐄 T∆V
dP⤩⤑⤰ dT ∆V is the volume change accompanying the phase change.
The latent heat of vaporization can be calculated from vapor-pressure and volumetric data.
Rough estimates of latent heats of vaporization for pure liquids at their normal boiling points:
∆⢒ ⤂⤄ ᕂ 10^ (Trouton’s rule)
2.12 Energy balance for open systems,
An open system involves mass flow in the energy balance. If mass flows in and out of a system, the mass carries energy (U, PE and KE) along with it.
∆E 㐄 Q ㎗ W ㎗ enegy input associated with mass )low in ㎘ energy output associated with mass )low out ㎗ )low work input ㎘ )low work output
Where ∆E is accumulation rate in energy in the system. It is zero at the steady state.
Energy input associated with mass flow in = ( U 1 + KE 1 + PE 1 )m 1 where U 1 , KE 1 , and PE 1 are the internal energy, kinetic energy, and potential energy, respectively, that are put in the system per unit mass, and m 1 is the mass of flow in.
Energy input associated with mass flow out = ( U 2 + KE 2 + PE 2 )m 2 where U 2 , KE 2 , and PE 2 are the internal energy, kinetic energy, and potential energy, respectively, that leave the system per unit mass, and m 2 is the mass of flow out.
Flow work input = p 1 V 1 m 1 where p 1 is the pressure and V 1 is the specific volume at the inlet.
Flow work output = p 2 V 2 m 2 where p 2 is the pressure and V 2 is the specific volume at the outlet.
∆E 㐄 Q ㎗ W ㎗ 䙦ផ⡩ ㎗ ច❸ ㎗ ថ❸䙧m⡩ ㎘ 䙦ផ❹ ㎗ ច❹ ㎗ ថ❹䙧m⡰ ㎗ p⡩V⡩m⡩ ㎘ p⡰V⡰m⡰
By rearranging,
∆E 㐄 Q ㎗ W ㎗ 䙦U⡩ ㎗ p⡩V⡩ ㎗ KE⡩ ㎗ PE⡩䙧 ㎘ 䙦U⡰ ㎗ p⡰V⡰ ㎗ KE⡰ ㎗ PE⡰䙧 㐄 Q ㎗ W ㎗ 䙦H⡩ ㎗ KE⡩ ㎗ PE⡩䙧 ㎘ 䙦H⡰ ㎗ KE⡰ ㎗ PE⡰䙧 㐄 Q ㎗ W ㎘ ∆䙦H ㎗ KE ㎗ PE䙧 This is equivalent to equation (2.28).
Example 2.
Example 2.
By definition, ∆H⤨⡨^ 㐄 ∑ υ⤙ H⤙
Q 㐄 㔳 n⤙,⡨ 㔅 C⤀,⤙⡨
⤄
⤄ㅧ,ㄖ
dT ㎗ ξ∆H⤨⡨
Q 㐄 㔳 n⤙,⡨ 㔅 C⤀,⤙⡨
⤄
⤄ㅧ,ㄖ
dT ㎗ ξ䙦∆H⤨⡨䙦25⤥C䙧 ㎗ 㔳 㔅 υ⤙C⤀,⤙⡨^ dT䙧
⤄
⡰⡷⡶.⡩⡳
All species involved in the reaction are counted.
The right hand side is a function of T for a given ξ.
Q=0 (adiabatic): no heat transfer across the boundary or completely insulated.
The temperature at the outlet with Q=0 is called adiabatic temperature
For a problem where T (outlet temperature) is given, Q can be determined. How much cooling is required for a specified outlet temperature?
For multiple reactions,
n⤙ 㐄 n⤙,⡨ ㎗ 㔳 υ⤙,⤠ ξ⤠
Q 㐄 㔳 n⤙,⡨ 㔅 C⤀,⤙⡨
⤄
⤄ㅧ,ㄖ
dT ㎗ 㔳 ξ⤠ ∆H⤨,⤠⡨
Q 㐄 㔳 n⤙,⡨ 㔅 C⤀,⤙⡨
⤄
⤄ㅧ,ㄖ
dT ㎗ 㔳 ξ⤠ 䙦∆H⤨,⤠⡨^ 䙦25⤥C䙧 ㎗ 㔅 㔳 υ⤙,⤠ C⤀,⤙⡨^ dT䙧
⤄
⡰⡷⡶.⡩⡳
Heating value: The negative of the standard heat of combustion for a fuel such coal or oil. Higher (gross) heating value (HHV): heating value when the product water is liquid. Lower (net) heating value (LHV): heating value when the product water is vapor.
Example 4.
Example 4.
Example 4.
Example 4.
