Thermodynamics: An Engineering Approach
Thermodynamics: An Engineering Approach
8th Edition
ISBN: 9780073398174
Author: Yunus A. Cengel Dr., Michael A. Boles
Publisher: McGraw-Hill Education
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Chapter 4.5, Problem 130RP

A well-insulated 3-m × 4–m × 6-m room initially at 7°C is heated by the radiator of a steam heating system. The radiator has a volume of 15 L and is filled with superheated vapor at 200 kPa and 200°C. At this moment both the inlet and the exit valves to the radiator are closed. A 120-W fan is used to distribute the air in the room. The pressure of the steam is observed to drop to 100 kPa after 45 min as a result of heat transfer to the room. Assuming constant specific heats for air at room temperature, determine the average temperature of air in 45 min. Assume the air pressure in the room remains constant at 100 kPa.

FIGURE P4–136

Chapter 4.5, Problem 130RP, A well-insulated 3-m  4m  6-m room initially at 7C is heated by the radiator of a steam heating

Expert Solution & Answer
Check Mark
To determine

The average temperature of air in 45 min.

Answer to Problem 130RP

The average temperature of air in 45 min is 10.7°C_.

Explanation of Solution

Show the free body diagram of the well-insulated room.

Thermodynamics: An Engineering Approach, Chapter 4.5, Problem 130RP

Write the expression for the energy balance equation.

EinEout=ΔEsystem (I)

Here, the total energy entering the system is Ein, the total energy leaving the system is Eout, and the change in the total energy of the system is ΔEsystem.

Substitute 0 for Ein, Qout for Eout, and m(u1u2) for ΔEsystem in Equation (I).

Qout=ΔU=m(u2u1)=m(u1u2) (II)

Here, the mass is m, the initial specific internal energy is u1, and the final specific internal energy is u2.

From the Table (A-4 through A-6), obtain the value of specific volume, specific internal energy at the initial and the final states.

At initial pressure and temperature of 200kPa and 200°C.

v1=1.08049m3/kgu1=2654.6kJ/kg

At final pressure of 100kPa.

vf=0.01043uf=417.40vg=1.6941m3/kgufg=2088.2kJ/kg

Write the expression for final quality at the final state.

x2=v1vfvfg (III)

Here, the specific volume of saturated liquid is vf, the specific volume of saturated vapour is vg, and the specific volume change upon vaporization is vfg.

Write the expression for final specific internal energy of a well-insulated room.

u2=uf+x2ufg (IV)

Here, the specific internal energy of saturated liquid is uf and the specific internal energy change upon vaporization is ufg.

Write the expression for total mass of a well-insulated room.

m=V1v1 (V)

Here, the initial specific volume is v1 and the volume of tank is V1.

Write expression for the volume of the well-insulated room.

V=lbh (VI)

Write the expression of mass of air in a well-insulated room.

mair=P1V1RT1 (VII)

Here, the initial pressure is P1, the initial volume is V1, the initial temperature is T1, and gas constant is R.

Write the expression for amount of fan work done in 45 min.

Wfan,in=W˙fan,inΔt (VIII)

Conclusion:

Here, the specific volume (v2=v1).

Substitute 1.08049m3/kg for v2, 0.01043m3/kg for vf, 1.693057m3/kg for vfg in Equation (III).

x2=(1.08049m3/kg)(0.01043m3/kg)(1.693057m3/kg)=0.6376

Substitute 417.40 for uf, 0.6376 for x2, and 2088kJ/kg for ufg in Equation (IV).

u2=417.40+0.6376(2088kJ/kg)=1748.7kJ/kg

Substitute 0.015m3 for V1 and 1.08049m3/kg for v1 in Equation (V).

m=0.015m31.08049m3/kg=0.01388kg

Substitute 1748.7kJ/kg for u2, 2654.6kJ/kg for u1, and 0.01388kg for m in Equation (II).

Qout=(0.01388kg)(2654.61748.7)kJ/kg=12.58kJ

Substitute 3m for l, 4m for b and 6m for h in Equation (VI).

V=3m×4m×6m=72m3

Substitute 100kPa for P1, 72m3 for V1, 200K for T1, and 0.2870kPam3/kgK for l in Equation (VII).

mair=(100kPa)(72m3)(0.2870kPam3/kgK)(200K)=89.60kg

Substitute 0.120kJ/s for W˙fan,in and 45min for Δt in Equation (VIII).

Wfan,in=(0.120kJ/s)(40min)=(0.120kJ/s)(40min×(60s1min))=324kJ

For well-insulated room the energy balance equation.\

Substitute Qin+Wfan,in for Ein, Wb,out for Eout, and ΔU for ΔEsystem in Equation (I).

Qin+Wfan,inWb,out=ΔUQin+Wfan,in=ΔHmcP(T2T1) (IX)

Express the change in internal energy and boundary work into the constant pressure expansion and compression in Equation (IX).

(Q˙in+W˙fan,in)Δt=mcP,avg(T2T1)                              (X)

Here, the rate of heat transfer entering is Q˙in, the rate of work of fan entering is W˙fan,in, change in temperature is Δt, the mass is m, the constant average pressure is cP,avg, the initial temperature is T1, and the final temperature is T2.

Substitute 12.58kJ for Q˙in, 324kJ for W˙fan,in, 89.60kg for m, 1.005kJ/kg°C for cP,avg, and 7°C for T1 in Equation (X).

(12.58kJ)+(324kJ)=(89.60kg)(1.005kJ/kg°C)(T27°C)(336.58kJ)=(90.048kJ°C)(T27°C)T2=10.7°C

Therefore, the well-insulated room temperature rises from 7°Cto 10.7°C in 45 min time.

Thus, the average temperature of air in 45 min is 10.7°C_.

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Chapter 4 Solutions

Thermodynamics: An Engineering Approach

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Prob. 107RPCh. 4.5 - Consider a pistoncylinder device that contains 0.5...Ch. 4.5 - Air in the amount of 2 lbm is contained in a...Ch. 4.5 - Air is expanded in a polytropic process with n =...Ch. 4.5 - Nitrogen at 100 kPa and 25C in a rigid vessel is...Ch. 4.5 - Prob. 112RPCh. 4.5 - Prob. 113RPCh. 4.5 - Prob. 114RPCh. 4.5 - 4–115 A mass of 12 kg of saturated...Ch. 4.5 - Prob. 116RPCh. 4.5 - Prob. 117RPCh. 4.5 - Prob. 118RPCh. 4.5 - Prob. 119RPCh. 4.5 - Prob. 120RPCh. 4.5 - Prob. 121RPCh. 4.5 - Prob. 122RPCh. 4.5 - Prob. 123RPCh. 4.5 - Prob. 124RPCh. 4.5 - Prob. 125RPCh. 4.5 - Prob. 126RPCh. 4.5 - Prob. 127RPCh. 4.5 - Prob. 128RPCh. 4.5 - A well-insulated 3-m 4m 6-m room initially at 7C...Ch. 4.5 - Prob. 131RPCh. 4.5 - Prob. 133RPCh. 4.5 - Prob. 134RPCh. 4.5 - An insulated pistoncylinder device initially...Ch. 4.5 - Prob. 137RPCh. 4.5 - Prob. 138RPCh. 4.5 - A pistoncylinder device initially contains 0.35 kg...Ch. 4.5 - Prob. 140RPCh. 4.5 - 4–141 One kilogram of carbon dioxide is compressed...Ch. 4.5 - Prob. 142RPCh. 4.5 - Prob. 143RPCh. 4.5 - Prob. 144FEPCh. 4.5 - A 3-m3 rigid tank contains nitrogen gas at 500 kPa...Ch. 4.5 - Prob. 146FEPCh. 4.5 - A well-sealed room contains 60 kg of air at 200...Ch. 4.5 - Prob. 148FEPCh. 4.5 - A room contains 75 kg of air at 100 kPa and 15C....Ch. 4.5 - A pistoncylinder device contains 5 kg of air at...Ch. 4.5 - Prob. 151FEPCh. 4.5 - Prob. 152FEPCh. 4.5 - A 2-kW electric resistance heater submerged in 5...Ch. 4.5 - 1.5 kg of liquid water initially at 12C is to be...Ch. 4.5 - An ordinary egg with a mass of 0.1 kg and a...Ch. 4.5 - 4–156 An apple with an average mass of 0.18 kg and...Ch. 4.5 - A 6-pack of canned drinks is to be cooled from 18C...Ch. 4.5 - An ideal gas has a gas constant R = 0.3 kJ/kgK and...Ch. 4.5 - Prob. 159FEPCh. 4.5 - Prob. 161FEP
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