System Dynamics
System Dynamics
3rd Edition
ISBN: 9780073398068
Author: III William J. Palm
Publisher: MCG
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Chapter 7, Problem 7.51P
To determine

Dynamic model for the system.

Temperature T(t) as function of time.

Expert Solution & Answer
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Answer to Problem 7.51P

Dynamic model for the system is 388.34dTdt=(70°FT).

Temperature T(t) as function of time is e[3.912t388.34]+70°F.

Explanation of Solution

Given information:

The initial temperature of water is 120°F, the length of copper pipe is 6ft, inner radius of copper pipe is 1/4in, the outer radius of the pipe is 3/8in, the temperature of the surrounding air is 70°F, thermal conductivity of copper is 50lb/sec°F, the convection coefficient at the inner surface between water and pipe is 6lb/secft°F, the convection coefficient at the outer surface is 1.1lb/secft°F, specific heat of water is 25000ftlb/slug°F, the density of water is 1.94slug/ft3.

Write the expression for area of the inner cylinder.

Ai=2πriL        ....... (I)

Here, the area of the inner cylinder is Ai, the inner radius is ri, the length of the cylinder is L.

Write the expression for area of the outer cylinder.

Ao=2πroL        ....... (II)

Here, the area of the outer cylinder is Ao, the outer radius is ro.

Write the expression for total contact resistance.

R=Ri+Ro+Rc        ....... (III)

Here, total contact resistance is R ,the resistance between the inner cylinder surface and water is Ri the resistance between the outer cylinder surface and air is Ro the resistance offered by the cylinder is Rc.

Substitute 1hiAi for Ri, 1hoAo for Ro and ln(rori)2πLk for Rc in Equation (III)

R=1hiAi+1hoAo+ln(rori)2πLk        ....... (IV)

Here, the convection coefficient at the inner surface between water and pipe is hi, the convection coefficient at the outer surface is ho, thermal conductivity of copper is k.

Write the expression for conservation of energy.

mcpdTdt=1R(TTo)..... (V)

Here, the mass of the copper pipe is m, the specific heat of water at constant volume is cp, the change of temperature with respect to time is dTdt, the temperature of the surrounding air is To, temperature of cylinder at any instant is T.

Write the expression for mass of the water flowing inside the pipe.

m=ρV        ....... (VI)

Here, the density of water is ρ, the volume of the cylinder is V.

Write the expression for volume of the cylinder.

V=πri2L

Here, the volume of the cylinder is V.

Substitute πri2L for V in Equation (VI)

m=ρ(πri2L)        ....... (VII)

Write the temperature difference between the cylinder and the surrounding.

θ=T70°F        ....... (IX)

Here, the temperature difference is θ.

Differentiate Equation (IX) with respect to t

dθdt=dTdt.

Calculation:

Substitute 1/4ft for ri, 6ft for L in Equation (I).

Ai=2π(6ft)(14in)=2π(6ft)(14in(0.083ft1in))=2π(6ft)(0.0207ft)=0.78ft2

Substitute 3/8ft for ro, 6ft for L in Equation (II).

Ao=2π(6ft)(38in)=2π(6ft)(38in(0.083ft1in))=2π(6ft)(0.0311ft)=1.172ft2

Substitute 1.172ft2 for Ao, 0.78ft2 for Ai, 50lb/sec°F for k, 6lb/secft°F for hi, 1.1lb/secft°F for ho, 1/4ft for ri, 3/8ft for ro and 6ft for L in Equation (IV).

R=[1(6lb/secft°F)(0.78ft2)+1(1.1lb/secft°F)(1.172ft2)+ln(3/8in1/4in)2π(6ft)(50lb/sec°F)]=[14.68lbft/sec°F+11.2892lbft/sec°F+ln(128)1884.9ftlb/sec°F]=[14.68lbft/sec°F+11.2892lbft/sec°F+0.4051884.9ftlb/sec°F]=0.2136+0.7756+2.15×104=0.9894sec°F/ftlb

Substitute 1.94slug/ft3 for ρ, 1/4in for ri and 6ft for L in Equation (VII).

m=(1.94slug/ft3)(π(1/4in)6ft)=(1.94slug/ft3)π((14)in(0.083ft1in))26ft=(1.94slug/ft3)(8.11×103ft3)=0.0157slug

Substitute 0.0157slug for m, 25000ftlb/slug°F for cp, 0.9894sec°F/ftlb for R and 70°F for To in Equation (V).

(0.0157slug)(25000ftlb/slug°F)dTdt=1(0.9894sec°F/ftlb)(T70°F)392.5ftlb/°FdTdt=1.0107ftlb/sec°F(T70°F)392.5ftlb/°F1.0107ftlb/sec°FdTdt=(70°FT)

388.34dTdt=(70°FT)        ....... (X)

Substitute θi for θ, Ti for T in Equation (IX).

θi=Ti70°F

Substitute 120°F for Ti.

θi=120°F70°F=50°F        ....... (XI)

Substitute T70°F for θ, dθdt for dTdt in Equation (X).

388.34dθdt=θ1θdθ=1388.34dt

Integrate both sides of the equation.

0t1θdθ=0t1388.34dtlnθlnθi=t388.34        ....... (XII)

Substitute T70 for θ and 50°F for θi in Equation (XII).

ln(T70°F)ln50°F=t388.34ln(T70°F)=t388.34+ln50°Fln(T70°F)=t388.34+3.912T(t)=e[3.912t388.34]+70°F.

Conclusion:

Dynamic model for the system is 388.34dTdt=(70°FT).

Temperature T(t) as function of time is e[3.912t388.34]+70°F.

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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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