System Dynamics
System Dynamics
3rd Edition
ISBN: 9780073398068
Author: III William J. Palm
Publisher: MCG
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Chapter 7, Problem 7.50P
To determine

(a)

The heat loss rate from the water to the air in the radial direction.

Expert Solution
Check Mark

Answer to Problem 7.50P

The heat loss rate from the water to the air in the radial direction is 195.3lbft/sec.

Explanation of Solution

Calculation:

Write the radial resistance of the pipe.

Rp=ln(rori)2πkL...... (I)

Here, the inner radius of the pipe is ri, the outer radius of the pipe is ro, the thermal conductivity of iron is k and the length of the pipe L.

Write the inner suface area of the pipe.

Ai=2πriL...... (II)

Write the inner convective resistance of the pipe.

Ri=1hiAi...... (III)

Here, the inner surface convective coefficient is hi.

Write the outer suface area of the pipe.

Ao=2πroL...... (IV)

Write the outer convective resistance of the pipe.

Ro=1hoAo...... (V)

Here, the outer surface convective coefficient is ho.

Write the total resistance of the pipe.

R=Ri+Rp+Ro...... (VI)

Calculate the heat loss rate from the water to the air.

q=(TwTo)R...... (VII)

Here, the temperature of the water is Tw and the temperature of the surrounding air is To.

Substitute 10ft for L, 12in for ri, 34in for ro, and 10.1lb/sec°F for k in Equation (I).

Rp=ln((34in)(12in))2π(10.1lb/sec°F)(10ft)=0.4052π(10.1lb/sec°F)(10ft)=6.389×104sec°F/lbft6.4×104sec°F/lbft

Substitute 10ft for L, and 12in for ri in Equation (II).

Ai=2π(12in)(10ft)=2π(12in)(1ft12in)(10ft)=(0.2617ft)(10ft)2.62ft2

Substitute 2.62ft2 for Ai and 16lb/ft-sec-°F for hi in Equation (III).

Ri=1(16lb/ftsec°F)(2.62ft2)=1(41.92lbft/sec°F)=0.0238sec°F/lbft0.024sec°F/lbft

Substitute 10ft for L, and 34in for ro in Equation (IV).

Ao=2π(34in)(10ft)=2π(34in)(1ft12in)(10ft)=3.926ft23.93ft2

Substitute 3.93ft2 for Ao and 1.1lb/ftsec°F for ho in Equation (V).

Ro=1(1.1lb/ftsec°F)(3.93ft2)=1(4.323lbft/sec°F)=0.231sec°F/lbft

Substitute 0.231sec°F/lbft for Ro, 6.4×104sec°F/lbft for Rp and 0.024sec°F/lbft for Ri in Equation (VI).

R=(0.024sec°F/lbft)+(6.4×104sec°F/lbft)+(0.231sec°F/lbft)=(0.024sec°F/lbft)+(0.23164sec°F/lbft)=0.25564sec°F/lbft0.256sec°F/lbft

Substitute 0.256sec°F/lbft for R

120°F for Tw and 70°F for To in Equation (VII).

q=(120°F70°F)(0.256sec°F/lbft)=(50°F)(0.256sec°F/lbft)195.3lbft/sec.

Conclusion:

Thus, the heat loss rate from the water to the air in the radial direction is 195.3lbft/sec.

To determine

(b)

The validity of the constant-temperature assumption.

Expert Solution
Check Mark

Answer to Problem 7.50P

The constant-temperature assumption is valid.

Explanation of Solution

Calculation:

Write the mass flow of the water.

m˙=ρAiV...... (VIII)

Here, the densityof the water is ρ and the flow velocity of the water is V.

Write the thermal energy of the water in the pipe.

E=m˙cpTw...... (IX)

Here, the specific heat at constant pressure of the water is cp.

Substitute 1.94slug/ft3 for ρ, 2.62ft2 for Ai, and 0.5ft/s for V in Equation (VIII).

m˙=(1.94slug/ft3)(2.62ft2)(0.5ft/s)=(1.94slug/ft3)(1.31ft3/s)2.54slug/s

Substitute 2.54slug/s for m˙, 25000lbft/slug°F for cp and 120°F for Tw in Equation (IX).

E=(2.54slug/s)(25000lbft/slug°F)(120°F)=(2.54slug/s)(3000000lbft/slug)=7.62×106lbft/s

The thermal energy of the water in the pipe is 7.62×106lbft/s.

Since the thermal energy of the water in the pipe is greater than the the heat loss rate from the water to the air in the radial direction, so the water will loose very less amount of heat.

Conclusion:

Thus, the constant-temperature assumption is valid.

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Chapter 7 Solutions

System Dynamics

Ch. 7 - 7.11 Derive the expression for the capacitance of...Ch. 7 - Air flows in a certain cylindrical pipe 1 m long...Ch. 7 - Derive the expression for the linearized...Ch. 7 - Consider the cylindrical container treated in...Ch. 7 - A certain tank has a bottom area A = 20 m2. The...Ch. 7 - A certain tank has a circular bottom area A = 20...Ch. 7 - The water inflow rate to a certain tank was kept...Ch. 7 - Prob. 7.18PCh. 7 - Prob. 7.19PCh. 7 - In the liquid level system shown in Figure P7.20,...Ch. 7 - The water height in a certain tank was measured at...Ch. 7 - Derive the model for the system shown in Figure...Ch. 7 - (a) Develop a model of the two liquid heights in...Ch. 7 - Prob. 7.24PCh. 7 - Design a piston-type damper using an oil with a...Ch. 7 - Prob. 7.26PCh. 7 - 7.27 An electric motor is sometimes used to move...Ch. 7 - Prob. 7.28PCh. 7 - Prob. 7.29PCh. 7 - Figure P7.3O shows an example of a hydraulic...Ch. 7 - Prob. 7.31PCh. 7 - Prob. 7.32PCh. 7 - Prob. 7.33PCh. 7 - Prob. 7.34PCh. 7 - Prob. 7.35PCh. 7 - Prob. 7.36PCh. 7 - Prob. 7.37PCh. 7 - (a) Determine the capacitance of a spherical tank...Ch. 7 - Obtain the dynamic model of the liquid height It...Ch. 7 - Prob. 7.40PCh. 7 - Prob. 7.41PCh. 7 - Prob. 7.42PCh. 7 - Prob. 7.43PCh. 7 - Prob. 7.44PCh. 7 - Prob. 7.45PCh. 7 - The copper shaft shown in Figure P7.46 consists of...Ch. 7 - A certain radiator wall is made of copper with a...Ch. 7 - A particular house wall consists of three layers...Ch. 7 - A certain wall section is composed of a 12 in. by...Ch. 7 - Prob. 7.50PCh. 7 - Prob. 7.51PCh. 7 - A steel tank filled with water has a volume of...Ch. 7 - Prob. 7.53PCh. 7 - Prob. 7.54PCh. 7 - Prob. 7.55P
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