System Dynamics
System Dynamics
3rd Edition
ISBN: 9780073398068
Author: III William J. Palm
Publisher: MCG
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Chapter 7, Problem 7.55P
To determine

(a)

The model for the behavior of temperature T1 and T2.

Expert Solution
Check Mark

Answer to Problem 7.55P

The model for the behavior of temperature T1 is dT1dt=1C1[1R2(T2T1)1R1(T1T0)].

The model for the behavior of temperature T2 is dT2dt=1C2[1R2(T2T1)].

Explanation of Solution

Calculation:

Write the expression for the thermal capacitance of the first mass.

C1=dq1dT1dq1=C1dT1

Here, the thermal capacitance is of the first mass is C1, the small quantity of heat added to the first mass is dq1, and the small change in the temperature of the first mass is dT1.

Divide both side by dt in the above expression.

dq1dt=C1dT1dt

Write the expression for the thermal capacitance of the second mass.

C2=dq2dT2dq2=C2dT2

Here, the thermal capacitance is of the first mass is C2, the small quantity of heat added to the second mass is dq2, and the small change in the temperature of the second mass is dT2.

Divide both sides by dt.

dq2dt=C2dT2dt

Write the expression for the conservation of heat of the first mass.

dq1dt=q2q1.... (I)

Here, the heat inflow rate to the first mass from the second mass is q2 and the heat outflow rate from the first mass is q1.

Since the second mass is insulated from the three sides, so there is only heat transfer takes place between the first mass and second mass.

Write the expression for the conservation of heat of the first mass.

dq2dt=q2.... (II)

Substitute C1dT1dt for dq1dt in Equation (I).

C1dT1dt=q2q1.... (III)

Substitute C2dT2dt for dq2dt in Equation (II).

C2dT2dt=q2.... (IV)

Write the expression for the convective heat transfer rate from the first mass to the air.

q1=h1A(T1T0).... (V)

Here, the convective heat transfer coefficient is h1 and the area of the wall boundary is A.

Write the expression for the convective heat transfer rate from the second mass to the first mass.

q2=h2A(T2T1).... (VI)

Here, the convective heat transfer coefficient is h2.

Write the expression for the thermal resistance of the heating element.

R1=1h1Ah1A=1R1

Write the expression for the thermal resistance for the surrounding heat transfer.

R2=1h2Ah2A=1R2

Substitute 1R1 for h1A in Equation (V).

q1=1R1(T1T0)

Substitute 1R2 for h2A in Equation (VI).

q2=1R2(T2T1)

Substitute 1R1(T1T0) for q1 and 1R2(T2T1) for q2 in Equation (III).

C1dT1dt=1R2(T2T1)1R1(T1T0)dT1dt=1C1[1R2(T2T1)1R1(T1T0)]

Thus, the model for the behavior of temperature T1 is dT1dt=1C1[1R2(T2T1)1R1(T1T0)].

Substitute 1R2(T2T1) for q2 in Equation (IV).

C2dT2dt=1R2(T2T1)dT2dt=1C2[1R2(T2T1)]

Thus, the model for the behavior of temperature T2 is dT2dt=1C2[1R2(T2T1)].

To determine

(b)

The effect on the dynamic model of temperature if the thermal capacitance C2 is very small.

Expert Solution
Check Mark

Answer to Problem 7.55P

The effect on dynamic model of temperature if the thermal capacitance C2 is very small is C1dT1dt=1R1(T1T0).

Explanation of Solution

Calculation:

Write the expression for the model behavior of temperature T2.

dT2dt=1C2[1R2(T2T1)]C2dT2dt=[1R2(T2T1)].... (VII)

Write the expression for the model behavior of temperature T1.

dT1dt=1C1[1R2(T2T1)1R1(T1T0)]C1dT1dt=[1R2(T2T1)1R1(T1T0)].... (VIII)

Since the thermal capacitance C2 is very small, so its value is supposed to be zero.

C2=0

Substitute 0 for C2 in Equation (VII).

(0)dT2dt=[1R2(T2T1)][1R2(T2T1)]=0T2T1=0T2=T1

Substitute T1 for T2 in Equation (VIII).

C1dT1dt=[1R2(T1T1)1R1(T1T0)]C1dT1dt=[01R1(T1T0)]C1dT1dt=1R1(T1T0)

Thus, the effect on dynamic model of temperature if the thermal capacitance C2 is very small is C1dT1dt=1R1(T1T0).

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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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