The application of Eqs. (2.31) and (2.32) to the solution of practical problems requires enthalpy values. Because H is a state function, its values depend only on point conditions; once deter-mined, they may be tabulated for subsequent use for the same sets of conditions. To this end, Eq. (2.32) may be applied to laboratory processes designed for enthalpy measurements.
Figure 2.6: Flow calorimeter.
Constant temperature bath
Discharge
Supply
Section 1
Section 2 Applied
emf
Heater Valve
T2 P2
A simple flow calorimeter is illustrated schematically in Fig. 2.6. Its essential feature is an electric resistance heater immersed in a flowing fluid. The design provides for minimal velocity and elevation changes from section 1 to section 2, making kinetic- and potential- energy changes of the fluid negligible. With no shaft work entering the system, Eq. (2.32) reduces to ΔH = H2− H1= Q. The rate of heat transfer to the fluid is determined from the resistance of the heater and the current passing through it. In practice a number of details require careful attention, but in principle the operation of the flow calorimeter is simple. Measurements of the heat transfer rate and flow rate allow calculation of the change ΔH between sections 1 and 2.
For example, enthalpies of both liquid and vapor H2O are readily determined. The constant-temperature bath is filled with a mixture of crushed ice and water to maintain a tem-perature of 0°C. Liquid water is supplied to the apparatus, and the coil that carries it through the constant-temperature bath is long enough to bring it to an exit temperature of essentially 0°C. The temperature and pressure at section 2 are measured by suitable instruments. Values of the enthalpy of H2O for various conditions at section 2 are given by:
H 2 = H 1 + Q
where Q is the heat added per unit mass of water flowing.
The pressure may vary from run to run, but in the range encountered here it has a neg-ligible effect on the enthalpy of the entering water, and for practical purposes H1 is a con-stant. Absolute values of enthalpy, like absolute values of internal energy, are unknown. An arbitrary value may therefore be assigned to H1 as the basis for all other enthalpy values.
Setting H1 = 0 for liquid water at 0°C makes:
H = H + Q = 0 + Q = Q
2.9. Mass and Energy Balances for Open Systems 55 Enthalpy values may be tabulated for the temperatures and pressures existing at section 2 for a large number of runs. In addition, specific-volume measurements made for these same conditions may be added to the table, along with corresponding values of the internal energy calculated by Eq. (2.10), U = H − PV. In this way, tables of thermodynamic properties are compiled over the entire useful range of conditions. The most widely used such tabulation is for H2O and is known as the steam tables.12
The enthalpy may be taken as zero for some other state than liquid at 0°C. The choice is arbitrary. The equations of thermodynamics, such as Eqs. (2.31) and (2.32), apply to changes of state, for which the enthalpy differences are independent of the location of the zero point.
However, once an arbitrary zero point is selected for the enthalpy, an arbitrary choice cannot be made for the internal energy, because internal energy is related to enthalpy by Eq. (2.10).
Example 2.12
For the flow calorimeter just discussed, the following data are taken with water as the test fluid:
Flow rate = 4.15 g⋅ s −1 t 1 = 0° C t 2 = 300° C P 2 = 3 bar Rate of heat addition from resistance heater = 12,740 W
The water is completely vaporized in the process. Calculate the enthalpy of steam at 300°C and 3 bar based on H = 0 for liquid water at 0°C.
Solution 2.12
If Δz and Δu2 are negligible and if Ws and H1 are zero, then H2 = Q, and H 2 = 12,740 J⋅ s __________−1
4.15 g⋅ s −1 = 3070 J⋅ g −1 or 3070 kJ⋅ kg −1
Example 2.13
Air at 1 bar and 25°C enters a compressor at low velocity, discharges at 3 bar, and enters a nozzle in which it expands to a final velocity of 600 m·s−1 at the initial condi-tions of pressure and temperature. If the work of compression is 240 kJ per kilogram of air, how much heat must be removed during compression?
12Steam tables adequate for many purposes are given in Appendix E. The Chemistry WebBook of NIST includes a fluid properties calculator with which one can generate tables for water and some 75 other substances: http://
webbook.nist.gov/chemistry/fluid/
Solution 2.13
Because the air returns to its initial conditions of T and P, the overall process pro-duces no change in enthalpy of the air. Moreover, the potential-energy change of the air is presumed negligible. Neglecting also the initial kinetic energy of the air, we write Eq. (2.31) as:
ΔH + ____Δ u 2
2 + gΔz = 0 + ___ u 22
2 + 0 = Q + W s Then
Q = u ___22 2 − W s The kinetic-energy term is evaluated as follows:
1
__2 u 22 = __1
2 (600 m__
s )
2 = 180,000 m ___2
s 2 = 180,000 ___ m 2 s 2 · kg___
kg
= 180,000 N⋅m⋅ kg −1 = 180 kJ⋅ kg −1
Then
Q = 180 − 240 = − 60 kJ⋅ kg −1
Heat in the amount of 60 kJ must be removed per kilogram of air compressed.
Example 2.14
Water at 90°C is pumped from a storage tank at a rate of 3 L·s−1. The motor for the pump supplies work at a rate of 1.5 kJ·s−1. The water goes through a heat exchanger, giving up heat at a rate of 670 kJ·s−1, and is delivered to a second storage tank at an elevation 15 m above the first tank. What is the temperature of the water delivered to the second tank?
Solution 2.14
This is a steady-state, steady-flow process for which Eq. (2.31) applies. The initial and final velocities of water in the storage tanks are negligible, and the term Δu2/2
1 bar, 25˚C
3 bar Ws = 240 kJ·kg–1
1 bar, 25˚C Compressor
u = 600 m·s–1
u = “low”
Q = ? kJ·kg–1
2.9. Mass and Energy Balances for Open Systems 57 may be omitted. All remaining terms are expressed in units of kJ·kg−1. At 90°C the density of water is 0.965 kg·L−1 and the mass flow rate is:
m
˙
= (3) (0.965) = 2.895 kg⋅ s −1∆z = 15 m
Q = −670 kJ s−1· Ws = 1.5 kJ s−1·
m = 2.895 kg s−1·
For the heat exchanger,
Q = − 670 / 2.895 = − 231.4 kJ⋅ kg −1 For the shaft work of the pump,
W s = 1.5 / 2.895 = 0.52 kJ⋅ kg −1
If g is taken as the standard value of 9.8 m·s−2, the potential-energy term is:
gΔz = (9.8) (15) = 147 m 2 ⋅s −2 = 147 J⋅ kg −1 = 0.147 kJ⋅ kg −1 Equation (2.31) now yields:
ΔH = Q + W s − gΔz = − 231.4 + 0.52 − 0.15 = − 231.03 kJ⋅ kg −1 The steam-table value for the enthalpy of liquid water at 90°C is:
H 1 = 376.9 kJ⋅ kg −1
Thus,
ΔH = H 2 − H 1 = H 2 − 376.9 = − 231.0 and
H 2 = 376.9 − 231.0 = 145.9 kJ⋅ kg −1
The temperature of water having this enthalpy is found from the steam tables:
t = 34.83° C
In this example, Ws and gΔz are small compared with Q, and for practical purposes could be neglected.
Example 2.15
A steam turbine operates adiabatically with a power output of 4000 kW. Steam enters the turbine at 2100 kPa and 475°C. The exhaust is saturated steam at 10 kPa that enters a condenser, where it is condensed and cooled to 30°C. What is the mass flow rate of the steam, and at what rate must cooling water be supplied to the condenser, if the water enters at 15°C and is heated to 25°C?
Solution 2.15
The enthalpies of entering and exiting steam from the turbine are found from the steam tables:
H 1 = 3411.3 kJ⋅ kg −1 and H 2 = 2584.8 kJ⋅ kg −1
For a properly designed turbine, kinetic- and potential-energy changes are negligible, and for adiabatic operation Q = 0. Eq. (2.32) becomes simply Ws = ΔH.
Then W ∙ s = m
˙
(ΔH), and m˙
steam = ____ W˙
sΔH = _____________________− 4000 kJ⋅ s −1
( 2584.8 − 3411.3 ) kJ⋅ kg −1 = 4.840 kg⋅ s −1
For the condenser, the steam condensate leaving is subcooled water at 30°C, for which (from the steam tables) H3 = 125.7 kJ·kg−1. For the cooling water entering at 15°C and leaving at 25°C, the enthalpies are
H in = 62.9 kJ⋅ kg −1 and H out = 104.8 kJ⋅ kg −1 Equation (2.29) here reduces to
m 4.840
˙
steam( 125.7 − 2584.8 ) + m ( H 3 − H 2 ) + m˙
water ( H˙
water out − H ( 104.8 − 62.9 ) = 0 in ) = 0 Solution gives,m
˙
water = 284.1 kg⋅ s −12.11. Problems 59
2.10 SYNOPSIS
After studying this chapter, including the end-of-chapter problems, one should be able to:
∙ State and apply the first law of thermodynamics, making use of the appropriate sign conventions
∙ Explain and employ the concepts of internal energy, enthalpy, state function, equilib-rium, and reversible process
∙ Explain the differences between state functions and path-dependent quantities such as heat and work
∙ Calculate changes in state variables for a real process by substituting a hypothetical reversible process connecting the same states
∙ Relate changes in the internal energy and enthalpy of a substance to changes in temper-ature, with calculations based on the appropriate heat capacity
∙ Construct and apply mass and energy balances for open systems
2.11 PROBLEMS
2.1. A nonconducting container filled with 25 kg of water at 20°C is fitted with a stirrer, which is made to turn by gravity acting on a weight of mass 35 kg. The weight falls slowly through a distance of 5 m in driving the stirrer. Assuming that all work done on the weight is transferred to the water and that the local acceleration of gravity is 9.8 m·s−2, determine:
(a) The amount of work done on the water.
(b) The internal energy change of the water.
(c) The final temperature of the water, for which CP= 4.18 kJ·kg−1·°C−1.
(d) The amount of heat that must be removed from the water to return it to its initial temperature.
(e) The total energy change of the universe because of (1) the process of lowering the weight, (2) the process of cooling the water back to its initial temperature, and (3) both processes together.
2.2. Rework Prob. 2.1 for an insulated container that changes in temperature along with the water and has a heat capacity equivalent to 5 kg of water. Work the problem with:
(a) The water and container as the system. (b) The water alone as the system.
2.3. An egg, initially at rest, is dropped onto a concrete surface and breaks. With the egg treated as the system,
(a) What is the sign of W?
(b) What is the sign of ΔEP?
(c) What is ΔEK? (d) What is ΔUt?
(e) What is the sign of Q?
In modeling this process, assume the passage of sufficient time for the broken egg to return to its initial temperature. What is the origin of the heat transfer of part (e)?
2.4. An electric motor under steady load draws 9.7 amperes at 110 volts, delivering 1.25(hp) of mechanical energy. What is the rate of heat transfer from the motor, in kW?
2.5. An electric hand mixer draws 1.5 amperes at 110 volts. It is used to mix 1 kg of cookie dough for 5 minutes. After mixing, the temperature of the cookie dough is found to have increased by 5°C. If the heat capacity of the dough is 4.2 kJ⋅kg−1⋅K−1, what frac-tion of the electrical energy used by the mixer is converted to internal energy of the dough? Discuss the fate of the remainder of the energy.
2.6. One mole of gas in a closed system undergoes a four-step thermodynamic cycle. Use the data given in the following table to determine numerical values for the missing quantities indicated by question marks.
Step ΔUt/J Q/J W/J
12 −200 ? −6000
23 ? −3800 ?
34 ? −800 300
41 4700 ? ?
12341 ? ? −1400
2.7. Comment on the feasibility of cooling your kitchen in the summer by opening the door to the electrically powered refrigerator.
2.8. A tank containing 20 kg of water at 20°C is fitted with a stirrer that delivers work to the water at the rate of 0.25 kW. How long does it take for the temperature of the water to rise to 30°C if no heat is lost from the water? For water, CP = 4.18 kJ⋅kg−1⋅°C−1. 2.9. Heat in the amount of 7.5 kJ is added to a closed system while its internal energy
decreases by 12 kJ. How much energy is transferred as work? For a process causing the same change of state but for which the work is zero, how much heat is transferred?
2.10. A steel casting weighing 2 kg has an initial temperature of 500°C; 40 kg of water ini-tially at 25°C is contained in a perfectly insulated steel tank weighing 5 kg. The cast-ing is immersed in the water and the system is allowed to come to equilibrium. What is its final temperature? Ignore the effects of expansion or contraction, and assume constant specific heats of 4.18 kJ⋅kg−1⋅K−1 for water and 0.50 kJ⋅kg−1⋅K−1 for steel.
2.11. Problems 61 2.11. An incompressible fluid (ρ = constant) is contained in an insulated cylinder fitted
with a frictionless piston. Can energy as work be transferred to the fluid? What is the change in internal energy of the fluid when the pressure is increased from P1 to P2? 2.12. One kg of liquid water at 25°C, for which CP= 4.18 kJ·kg−1·°C−1:
(a) Experiences a temperature increase of 1 K. What is ΔUt, in kJ?
(b) Experiences a change in elevation Δz. The change in potential energy ΔEP is the same as ΔUt for part (a). What is Δz, in meters?
(c) Is accelerated from rest to final velocity u. The change in kinetic energy ΔEK is the same as ΔUt for part (a). What is u, in m·s−1?
Compare and discuss the results of the three preceding parts.
2.13. An electric motor runs “hot” under load, owing to internal irreversibilities. It has been suggested that the associated energy loss be minimized by thermally insulating the motor casing. Comment critically on this suggestion.
2.14. A hydroturbine operates with a head of 50 m of water. Inlet and outlet conduits are 2 m in diameter. Estimate the mechanical power developed by the turbine for an outlet velocity of 5 m⋅s−1.
2.15. A wind turbine with a rotor diameter of 40 m produces 90 kW of electrical power when the wind speed is 8 m⋅s−1. The density of air impinging on the turbine is 1.2 kg⋅m−3. What fraction of the kinetic energy of the wind impinging on the turbine is converted to electrical energy?
2.16. The battery in a laptop computer supplies 11.1 V and has a capacity of 56 W⋅h. In ordinary use, it is discharged after 4 hours. What is the average current drawn by the laptop, and what is the average rate of heat dissipation from it? You may assume that the temperature of the computer remains constant.
2.17. Suppose that the laptop of Prob. 2.16 is placed in an insulating briefcase with a fully charged battery, but it does not go into “sleep” mode, and the battery discharges as if the laptop were in use. If no heat leaves the briefcase, the heat capacity of the brief-case itself is negligible, and the laptop has a mass of 2.3 kg and an average specific heat of 0.8 kJ⋅kg−1⋅°C−1, estimate the temperature of the laptop after the battery has fully discharged.
2.18. In addition to heat and work flows, energy can be transferred as light, as in a photo-voltaic device (solar cell). The energy content of light depends on both its wavelength (color) and its intensity. When sunlight impinges on a solar cell, some is reflected, some is absorbed and converted to electrical work, and some is absorbed and con-verted to heat. Consider an array of solar cells with an area of 3 m2. The power of sunlight impinging upon it is 1 kW⋅m−2. The array converts 17% of the incident power to electrical work, and it reflects 20% of the incident light. At steady state, what is the rate of heat removal from the solar cell array?
2.19. Liquid water at 180°C and 1002.7 kPa has an internal energy (on an arbitrary scale) of 762.0 kJ⋅kg−1 and a specific volume of 1.128 cm3⋅g−1.
(a) What is its enthalpy?
(b) The water is brought to the vapor state at 300°C and 1500 kPa, where its internal energy is 2784.4 kJ⋅kg−1 and its specific volume is 169.7 cm3⋅g−1. Calculate ΔU and ΔH for the process.
2.20. A solid body at initial temperature T0 is immersed in a bath of water at initial temper-ature Tw0. Heat is transferred from the solid to the water at a rate Q
= K ⋅ ( T w – T) , where K is a constant and Tw and T are instantaneous values of the temperatures of the water and solid. Develop an expression for T as a function of time τ. Check your result for the limiting cases, τ = 0 and τ = ∞. Ignore effects of expansion or contraction, and assume constant specific heats for both water and solid.
2.21. A list of common unit operations follows:
(a) Single-pipe heat exchanger (b) Double-pipe heat exchanger (c) Pump
(d) Gas compressor (e) Gas turbine (f) Throttle valve (g) Nozzle
Develop a simplified form of the general steady-state energy balance appropriate for each operation. State carefully, and justify, any assumptions you make.
2.22. The Reynolds number Re is a dimensionless group that characterizes the intensity of a flow. For large Re, a flow is turbulent; for small Re, it is laminar. For pipe flow, Re ≡ uρD/μ, where D is pipe diameter and μ is dynamic viscosity.
(a) If D and μ are fixed, what is the effect of increasing mass flow rate m
˙
on Re?(b) If m
˙
and μ are fixed, what is the effect of increasing D on Re?2.23. An incompressible (ρ = constant) liquid flows steadily through a conduit of circular cross-section and increasing diameter. At location 1, the diameter is 2.5 cm and the velocity is 2 m⋅s−1; at location 2, the diameter is 5 cm.
(a) What is the velocity at location 2?
(b) What is the kinetic-energy change (J⋅kg−1) of the fluid between locations 1 and 2?
2.24. A stream of warm water is produced in a steady-flow mixing process by combining 1.0 kg⋅s−1 of cool water at 25°C with 0.8 kg⋅s−1 of hot water at 75°C. During mixing, heat is lost to the surroundings at the rate of 30 kJ⋅s−1. What is the temperature of the warm water stream? Assume the specific heat of water is constant at 4.18 kJ⋅kg−1⋅K−1.
2.11. Problems 63 2.25. Gas is bled from a tank. Neglecting heat transfer between the gas and the tank, show
that mass and energy balances produce the differential equation:
dU
________
H′ − U = ___dm m
Here, U and m refer to the gas remaining in the tank; H′ is the specific enthalpy of the gas leaving the tank. Under what conditions can one assume H′ = H?
2.26. Water at 28°C flows in a straight horizontal pipe in which there is no exchange of either heat or work with the surroundings. Its velocity is 14 m⋅s−1 in a pipe with an internal diameter of 2.5 cm until it flows into a section where the pipe diameter abruptly increases. What is the temperature change of the water if the downstream diameter is 3.8 cm? If it is 7.5 cm? What is the maximum temperature change for an enlargement in the pipe?
2.27. Fifty (50) kmol per hour of air is compressed from P1 = 1.2 bar to P2 = 6.0 bar in a steady-flow compressor. Delivered mechanical power is 98.8 kW. Temperatures and velocities are:
T1 = 300 K T2 = 520 K
u1 = 10 m⋅s−1 u2 = 3.5 m⋅s−1
Estimate the rate of heat transfer from the compressor. Assume for air that C P = 7_2 R and that enthalpy is independent of pressure.
2.28. Nitrogen flows at steady state through a horizontal, insulated pipe with inside diam-eter of 1.5(in). A pressure drop results from flow through a partially opened valve.
Just upstream from the valve the pressure is 100(psia), the temperature is 120(°F), and the average velocity is 20(ft)·s−1. If the pressure just downstream from the valve is 20(psia), what is the temperature? Assume for air that PV/ T is constant, CV = (5/2)R, and CP = (7/2)R. (Values for R, the ideal gas constant, are given in App. A.)
2.29. Air flows at steady state through a horizontal, insulated pipe with inside diameter of 4 cm. A pressure drop results from flow through a partially opened valve. Just upstream from the valve, the pressure is 7 bar, the temperature is 45°C, and the aver-age velocity is 20 m⋅s−1. If the pressure just downstream from the valve is 1.3 bar, what is the temperature? Assume for air that PV/T is constant, CV = (5/2)R, and CP = (7/2)R. (Values for R, the ideal gas constant, are given in App. A.)
2.30. Water flows through a horizontal coil heated from the outside by high-temperature flue gases. As it passes through the coil, the water changes state from liquid at 200 kPa and 80°C to vapor at 100 kPa and 125°C. Its entering velocity is 3 m⋅s−1 and its exit velocity is 200 m⋅s−1. Determine the heat transferred through the coil per unit mass of water. Enthalpies of the inlet and outlet streams are:
Inlet: 334.9 kJ⋅kg−1; Outlet: 2726.5 kJ⋅kg−1
2.31. Steam flows at steady state through a converging, insulated nozzle, 25 cm long and with an inlet diameter of 5 cm. At the nozzle entrance (state 1), the temperature and pressure are 325°C and 700 kPa and the velocity is 30 m⋅s−1. At the nozzle exit (state 2), the steam temperature and pressure are 240°C and 350 kPa. Property values are:
H1 = 3112.5 kJ⋅kg−1 V1 = 388.61 cm3⋅g−1 H2 = 2945.7 kJ⋅kg−1 V2 = 667.75 cm3⋅g−1
What is the velocity of the steam at the nozzle exit, and what is the exit diameter?
2.32. In the following take CV = 20.8 and CP = 29.1 J⋅mol−1⋅°C−1 for nitrogen gas:
(a) Three moles of nitrogen at 30°C, contained in a rigid vessel, is heated to 250°C. How much heat is required if the vessel has a negligible heat capacity? If the vessel weighs 100 kg and has a heat capacity of 0.5 kJ⋅kg−1⋅°C−1, how much heat is required?
(b) Four moles of nitrogen at 200°C is contained in a piston/cylinder arrangement.
(b) Four moles of nitrogen at 200°C is contained in a piston/cylinder arrangement.