3.14 Applying the Energy Balance Using
3.14.2 Using Constant Specific Heats
When the specific heats are taken as constants, Eqs. 3.40 and 3.43 reduce, respec- tively, to
u(T2)2u(T1)5c𝜐(T22T1) (3.50) h(T2)2h(T1)5cp(T22T1) (3.51) Equations 3.50 and 3.51 are often used for thermodynamic analyses involving ideal gases because they enable simple closed-form equations to be developed for many processes.
The constant values of c𝜐 and cp in Eqs. 3.50 and 3.51 are, strictly speaking, mean values calculated as follows:
c𝜐5
#TT2
1
c𝜐(T) dT T22T1
, cp5
#TT2
1
cp(T) dT T22T1
However, when the variation of c𝜐 or cp over a given temperature interval is slight, little error is normally introduced by taking the specific heat required by Eq. 3.50 or 3.51 as the arithmetic average of the specific heat values at the two end tem- peratures. Alternatively, the specific heat at the average temperature over the Analysis: An energy balance for the closed system is
DKE01 DPE01 DU5Q2W
where the kinetic and potential energy terms vanish by assumption 2. Solving for W
➌ W5Q2 DU5Q2m(u22u1)
From the problem statement, Q 5 220 kJ. Also, from Table A-22, at T1 5 300 K, u1 5 214.07 kJ/kg, and at T2 5 470 K, u2 5 337.32 kJ/kg. Accordingly
W5 2 20 kJ 2(0.9 kg)(337.322214.07)5 2130.9 kJ The minus sign indicates that work is done on the system in the process.
➊ Although the initial and final states are assumed to be equilibrium states, the intervening states are not necessarily equilibrium states, so the process has been indicated on the accompanying p–𝜐 diagram by a dashed line. This dashed line does not define a “path” for the process.
➋ Table A-1 gives pc 5 37.7 bars, Tc 5 133 K for air. Therefore, at state 1, pR1 5 0.03, TR1 5 2.26, and at state 2, pR2 5 0.16, TR2 5 3.51. Referring to Fig. A-1, we conclude that at these states Z ⬇ 1, as assumed in the solution.
➌ In principle, the work could be evaluated through 兰p dV, but because the variation of pressure at the piston face with volume is not known, the inte- gration cannot be performed without more information.
Replacing air by carbon dioxide, but keeping all other problem statement details the same, evaluate work, in kJ. Ans. 2131.99 kJ
Ability to…
❑ define a closed system and identify interactions on its boundary.
❑ apply the energy balance using the ideal gas model.
✓Skills Developed
3.14 Applying the Energy Balance Using Ideal Gas Tables, Constant Specific Heats, and Software 123
c c c c EXAMPLE 3.10 c
interval can be used. These methods are particularly convenient when tabular specific heat data are available, as in Tables A-20, for then the constant specific heat values often can be determined by inspection.
assuming the specific heat cʋ is a constant and using Eq. 3.50, the expression for work in the solution of Example 3.9 reads
W5Q2mc𝜐(T22T1)
Evaluating cʋ at the average temperature, 383 K (110°C), Table A-20 gives cʋ 5 0.721 kJ/kg ? K. Inserting this value for cʋ together with other data from Example 3.9
W5 220 kJ2 (0.9 kg)(0.721 kJ/kg?K)(4702300) K 5 2130.31 kJ
which agrees closely with the answer obtained in Example 3.9 using Table A-22 data.b b b b b
The following example illustrates the use of the closed-system energy balance, together with the ideal gas model and the assumption of constant specific heats.
Using the Energy Balance and Constant Specific Heats
Two tanks are connected by a valve. One tank contains 2 kg of carbon monoxide gas at 77°C and 0.7 bar. The other tank holds 8 kg of the same gas at 27°C and 1.2 bar. The valve is opened and the gases are allowed to mix while receiving energy by heat transfer from the surroundings. The final equilibrium temperature is 42°C. Using the ideal gas model with constant cʋ, determine (a) the final equilibrium pressure, in bar, (b) the heat transfer for the process, in kJ.
SOLUTION
Known: Two tanks containing different amounts of carbon monoxide gas at initially different states are connected by a valve. The valve is opened and the gas allowed to mix while receiving energy by heat transfer. The final equilibrium temperature is known.
Find: Determine the final pressure and the heat transfer for the process.
Schematic and Given Data:
Analysis:
(a) The final equilibrium pressure pf can be determined from the ideal gas equation of state pf5 mRTf
V Engineering Model:
1. The total amount of carbon monoxide gas is a closed system.
2. The gas is modeled as an ideal gas with constant c𝜐.
3. The gas initially in each tank is in equilibrium. The final state is an equilibrium state.
4. No energy is transferred to, or from, the gas by work.
5. There is no change in kinetic or potential energy.
Fig. E3.10
Carbon monoxide
Tank 1 Tank 2
2 kg, 77°C, 0.7 bar
Carbon monoxide 8 kg, 27°C,
1.2 bar
Valve ➊
124 Chapter 3 Evaluating Properties
where m is the sum of the initial amounts of mass present in the two tanks, V is the total volume of the two tanks, and Tf is the final equilibrium temperature. Thus
pf5 (m11m2)RTf
V11V2
Denoting the initial temperature and pressure in tank 1 as T1 and p1, respectively, V1 5 m1RT1/p1. Similarly, if the initial temperature and pressure in tank 2 are T2 and p2, V2 5 m2RT2/p2. Thus, the final pressure is
pf5 (m11m2)RTf
am1RT1
p1 b1am2RT2
p2 b
5 (m11m2)Tf
am1T1
p1 b1am2T2
p2 b Inserting values
pf5 (10 kg)(315 K) (2 kg)(350 K)
0.7 bar 1 (8 kg)(300 K) 1.2 bar
51.05 bar
(b) The heat transfer can be found from an energy balance, which reduces with assumptions 4 and 5 to give DU5Q2W0
or
Q5Uf2Ui
Ui is the initial internal energy, given by
Ui5m1u(T1)1m2u(T2)
where T1 and T2 are the initial temperatures of the CO in tanks 1 and 2, respectively. The final internal energy is Uf
Uf5(m11m2)u(Tf)
Introducing these expressions for internal energy, the energy balance becomes Q5m1[u(Tf)2u(T1)]1m2[u(Tf)2u(T2)]
Since the specific heat c𝜐 is constant (assumption 2)
Q5m1c𝜐(Tf2T1)1m2c𝜐(Tf2T2)
Evaluating c𝜐 as the average of the values listed in Table A-20 at 300 K and 350 K, c𝜐 5 0.745 kJ/kgⴢK. Hence Q5(2 kg)a0.745 kJ
kg?Kb(315 K2350 K) 1(8 kg)a0.745 kJ
kg?Kb(315 K2300 K) 5 +37.25 kJ
The plus sign indicates that the heat transfer is into the system.
➊ By referring to a generalized compressibility chart, it can be verified that the ideal gas equation of state is appropriate for CO in this range of temperature and pressure. Since the specific heat c𝜐 of CO varies little over the temperature interval from 300 to 350 K (Table A-20), it can be treated as constant with acceptable accuracy.
)
Ability to…
❑ define a closed system and identify interactions on its boundary.
❑ apply the energy balance using the ideal gas model when the specific heat c𝜐 is constant.
✓Skills Developed
Evaluate Q using specific internal energy values for CO from Table A-23. Compare with the result using constant c𝜐. Ans. 36.99 kJ
3.14 Applying the Energy Balance Using Ideal Gas Tables, Constant Specific Heats, and Software 125