PROBLEMS - The First Law and Other Basic Concepts

The First Law and Other Basic Concepts

2.11 PROBLEMS

(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 fraction 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 ΔU^t/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, C_P = 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 initially at 25°C is contained in a perfectly insulated steel tank weighing 5 kg. The casting 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 C_P= 4.18 kJ·kg⁻¹·°C⁻¹:

(a) Experiences a temperature increase of 1 K. What is ΔU^t, in kJ?

(b) Experiences a change in elevation Δz. The change in potential energy ΔE_P is the same as ΔU^t for part (a). What is Δz, in meters?

(c) Is accelerated from rest to final velocity u. The change in kinetic energy ΔE_K is the same as ΔU^t 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 briefcase 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 photovol- taic 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 converted 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 temperature T_w₀. Heat is transferred from the solid to the water at a rate Q

= K ⋅ ( T w – T) , where K is a constant and T_w 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

(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:

________

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 an inside diameter 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, C_V = (5/2)R, and C_P = (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 an 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 average 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, C_V = (5/2)R, and C_P = (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 C_V = 20.8 and C_P = 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.

How much heat must be extracted from this system, which is kept at constant pressure, to cool it to 40°C if the heat capacity of the piston and cylinder is neglected?

2.33. In the following take C_V = 5 and C_P = 7(Btu)(lb mole)⁻¹(°F)⁻¹ for nitrogen gas:

(a) Three pound moles of nitrogen at 70(°F), contained in a rigid vessel, is heated to 350(°F). How much heat is required if the vessel has a negligible heat capacity?

If it weighs 200(lbm) and has a heat capacity of 0.12(Btu)(lbm)⁻¹(°F)⁻¹, how much heat is required?

(b) Four pound moles of nitrogen at 400(°F) is contained in a piston/cylinder arrangement.

How much heat must be extracted from this system, which is kept at constant pressure, to cool it to 150(°F) if the heat capacity of the piston and cylinder is neglected?

2.34. Find an equation for the work of reversible, isothermal compression of 1 mol of gas in a piston/cylinder assembly if the molar volume of the gas is given by

V = _{___}RT

P + b where b and R are positive constants.

2.35. Steam at 200(psia) and 600(°F) [state 1] enters a turbine through a 3 inch diameter pipe with a velocity of 10(ft)⋅s⁻¹. The exhaust from the turbine is carried through a 10 inch diameter pipe and is at 5(psia) and 200(°F) [state 2]. What is the power output of the turbine?

H₁ = 1322.6(Btu)(lbm)⁻¹ V₁ = 3.058(ft)³(lbm)⁻¹ H₂ = 1148.6(Btu)(lbm)⁻¹ V₂ = 78.14(ft)³(lbm)⁻¹

2.36. Steam at 1400 kPa and 350°C [state 1] enters a turbine through a pipe that is 8 cm in diameter, at a mass flow rate of 0.1 kg⋅s⁻¹. The exhaust from the turbine is carried through a 25 cm diameter pipe and is at 50 kPa and 100°C [state 2]. What is the power output of the turbine?

2.11. Problems 65

2.37. Carbon dioxide gas enters a water-cooled compressor at conditions P₁ = 1 bar and T₁ = 10°C, and is discharged at conditions P₂ = 36 bar and T₂ = 90°C. The entering CO2 flows through a 10 cm diameter pipe with an average velocity of 10 m⋅s⁻¹, and is discharged through a 3 cm diameter pipe. The power supplied to the compressor is 12.5 kJ·mol⁻¹. What is the heat-transfer rate from the compressor?

H₁ = 21.71 kJ⋅mol⁻¹ V₁ = 23.40 L⋅mol⁻¹ H₂ = 23.78 kJ⋅mol⁻¹ V₂ = 0.7587 L⋅mol⁻¹

2.38. Carbon dioxide gas enters a water-cooled compressor at conditions P₁ = 15(psia) and T₁ = 50(°F), and is discharged at conditions P₂ = 520(psia) and T₂ = 200(°F). The entering CO2 flows through a 4 inch diameter pipe with a velocity of 20(ft)⋅s⁻¹, and is discharged through a 1 inch diameter pipe. The shaft work supplied to the compressor is 5360(Btu)(lb mole)⁻¹. What is the heat-transfer rate from the compressor in (Btu)·h⁻¹?

H₁ = 307(Btu)(lbm)⁻¹ V₁ = 9.25(ft)³(lbm)⁻¹ H₂ = 330(Btu)(lbm)⁻¹ V₂ = 0.28(ft)³(lbm)⁻¹

2.39. Show that W and Q for an arbitrary mechanically reversible nonflow process are given by:

W = ∫ V dp − Δ(PV ) Q = ΔH − ∫ V dp

2.40. One kilogram of air is heated reversibly at constant pressure from an initial state of 300 K and 1 bar until its volume triples. Calculate W, Q, ΔU, and ΔH for the process.

Assume for air that PV / T = 83.14 bar⋅cm³⋅mol⁻¹⋅K⁻¹ and C_P = 29 J⋅mol⁻¹⋅K⁻¹. 2.41. The conditions of a gas change in a steady-flow process from 20°C and 1000 kPa

to 60°C and 100 kPa. Devise a reversible nonflow process (any number of steps) for accomplishing this change of state, and calculate ΔU and ΔH for the process on the basis of 1 mol of gas. Assume for the gas that PV/T is constant, C_V = (5/2)R, and C_P = (7/2)R.

2.42. A flow calorimeter like that shown in Fig. 2.6 is used with a flow rate of 20 g⋅min⁻¹ of the fluid being tested and a constant temperature of 0°C leaving the constant- temperature bath. The steady-state temperature at section two (T₂) is measured as a function of the power supplied to the heater (P), to obtain the data shown in the table below. What is the average specific heat of the substance tested over the temperature range from 0°C to 10°C? What is the average specific heat from 90°C to 100°C? What is the average specific heat over the entire range tested? Describe how you would use this data to derive an expression for the specific heat as a function of temperature.

H₁ = 3150.7 kJ⋅kg⁻¹ V₁ = 0.20024 m³⋅kg⁻¹ H₂ = 2682.6 kJ⋅kg⁻¹ V₂ = 3.4181 m³⋅kg⁻¹

2.43. Like the flow calorimeter of Fig. 2.6, a particular single-cup coffee maker uses an electric heating element to heat a steady flow of water from 22°C to 88°C. It heats 8 fluid ounces of water (with a mass of 237 g) in 60 s. Estimate the power requirement of the heater during this process. You may assume the specific heat of water is constant at 4.18 J⋅g⁻¹⋅°C⁻¹.

2.44. (a) An incompressible fluid (ρ = constant) flows through a pipe of constant cross- sectional area. If the flow is steady, show that velocity u and volumetric flow rate q are constant.

(b) A chemically reactive gas stream flows steadily through a pipe of constant cross-sectional area. Temperature and pressure vary with pipe length. Which of the following quantities are necessarily constant: m

˙

^,ⁿ

˙

^,^q^,^u^?

2.45. The mechanical-energy balance provides a basis for estimating pressure drop owing to friction in fluid flow. For steady flow of an incompressible fluid in a horizontal pipe of constant cross-sectional area, it may be written,

ΔP

___ΔL + _{__}2

D f F ρu ² = 0

where f_F is the Fanning friction factor. Churchill¹⁴ gives the following expression for f_F for turbulent flow:

f F = 0.3305 { ln [ 0.27 ^__D^∈ + ( ^__Re⁷ ) ^0.9 ] }⁻²

Here, Re is the Reynolds number and ∈ /D is the dimensionless pipe roughness. For pipe flow, Re ≡ uρD/μ, where D is pipe diameter and μ is dynamic viscosity. The flow is turbulent for Re > 3000.

Consider the flow of liquid water at 25°C. For one of the sets of conditions given below, determine m

˙

(in kg⋅s⁻¹) and ΔP/ΔL (in kPa⋅m⁻¹). Assume ∈ /D = 0.0001. For liquid water at 25°C, ρ = 996 kg⋅m⁻³, and μ = 9.0 × 10⁻⁴ kg⋅m⁻¹⋅s⁻¹. Verify that the flow is turbulent.

(a) D = 2 cm, u = 1 m·s⁻¹ (b) D = 5 cm, u = 1 m·s⁻¹ (c) D = 2 cm, u = 5 m·s⁻¹ (d) D = 5 cm, u = 5 m·s⁻¹

2.46. Ethylene enters a turbine at 10 bar and 450 K, and exhausts at 1(atm) and 325 K. For m

˙

= 4.5 kg⋅s⁻¹, determine the cost C of the turbine. State any assumptions you make.

Data : H 1 = 761.1 H 2 = 536.9 kJ⋅ kg ⁻¹ C / $ = (15,200) (

|

^˙

|

^{/ kW}⁾^0.573

14AIChE J., vol. 19, pp. 375–376, 1973.

T₂/°C 10 20 30 40 50 60 70 80 90 100

P/W 5.5 11.0 16.6 22.3 28.0 33.7 39.6 45.4 51.3 57.3

2.11. Problems 67 2.47. The heating of a home to increase its temperature must be modeled as an open system

because expansion of the household air at constant pressure results in leakage of air to the outdoors. Assuming that the molar properties of air leaving the home are the same as those of the air in the home, show that energy and mole balances yield the following differential equation:

Q˙ = –PV dn_{___}

dt + n _{___}dU dt

Here, Q˙ is the rate of heat transfer to the air in the home, and t is time. Quantities P, V, n, and U refer to the air in the home.

2.48. (a) Water flows through the nozzle of a garden hose. Find an expression for m˙ in terms of line pressure P₁, ambient pressure P₂, inside hose diameter D₁, and nozzle outlet diameter D₂. Assume steady flow, and isothermal, adiabatic operation. For liquid water modeled as an incompressible fluid, H₂ − H₁ = (P₂ − P₁)/ρ for constant temperature.

(b) In fact, the flow cannot be truly isothermal: we expect T₂ > T₁, owing to fluid friction. Hence, H₂ − H₁ = C(T₂ − T₁) + (P₂ − P₁)/ρ, where C is the specific heat of water. Directionally, how would inclusion of the temperature change affect the value of m

˙

as found in part (a)?

2.49 Consider the process of boiling water to make pasta. A particular cylindrical pasta pot has a diameter of 24 cm and is initially filled to a depth of 10 cm with water at 20°C.

The water is brought to a boil, and just before adding pasta, the water level in the pot has decreased to 9.5 cm because some of the water has left as steam. Using data from the steam tables (App. E), estimate the following:

(a) The initial mass of water in the pot.

(b) The mass of water remaining in the pot just before pasta is added and the mass lost as vapor.

For simplicity, consider the water lost as vapor to be at 100°C.

(d) The total amount of heat transferred to the water.

2.50 Consider a microwave that is rated 1100 W and that at full power heats 250 mL of water from 20°C to 95°C in 2 minutes.

(a) What is the total change in internal energy of the water?

(b) If the electrical power supplied to the microwave is really 1100 W, what is the total quantity of electrical work supplied to the microwave in 2 minutes?

Chapter 3 Volumetric Properties

Dalam dokumen Introduction to Chemical Engineering Thermodynamics (Halaman 78-87)