Refer to Figure 7.1.4a, and suppose that p1 p2 = 10

For the hydraulic system shown in Figure P7.1, given \(A_{1}=10 \text { in. }^{2}, A_{2}=30 \text { in. }^{2}\), and mg = 60 lb, find the force \(f_{1}\) required to lift the mass m a distance \(x_{2}=6 \mathrm{in}\). Also find the distance \(x_{1}\) and the work done by the force \(f_{1}\).

Refer to the water storage and supply system shown in Figure 7.1.2. The cylindrical tank has a radius of 11 ft, and the water height is initially 5 ft. Find the water height after 5 hr if 1000 gallons per minute are pumped out of the well and 800 gallons per minute are withdrawn from the tank. Note that 1 gallon is \(0.13368 \ \mathrm{ft}^{3}\).

Consider the piston and mass shown in Figure 7.1.4a. Suppose there is dry friction acting between the mass m and the surface. Find the minimum area A of the piston required to move the mass against the friction force \(\mu m g\), where \(\mu=0.6\), mg = 1000 N, \(p_{1}=3 \times 10^{5} \mathrm{~Pa}\), and \(p_{2}=10^{5} \mathrm{~Pa}\).

In Figure P7.4 the piston of area A is connected to the axle of the cylinder of radius R, mass m, and inertia I about its center. Given \(p_{1}-p_{2}=3 \times 10^{5} \mathrm{~Pa}\), \(A=0.005 \mathrm{~m}^{2}\), R = 0.4 m, m = 100 kg, and \(I=7 \mathrm{~kg} \cdot \mathrm{m}^{2}\), determine the angular velocity \(\omega(t)\) of the cylinder assuming that it starts from rest.

Refer to Figure 7.1.4a, and suppose that \(p_{1}-p_{2}=10 \ \mathrm{lb} / \mathrm{in}^{2}, A=3 \text { in. }^{2}\), and mg = 600 lb. If the mass starts from rest at x(0) = 0, how far will it move in 0.5 sec, and how much hydraulic fluid will be displaced?

Pure water flows into a mixing tank of volume \(V=300 \mathrm{~m}^{3}\) at the constant volume rate of \(10 \mathrm{~m}^{3} / \mathrm{s}\). A solution with a salt concentration of \(s_{i} \mathrm{~kg} / \mathrm{m}^{3}\) flows into the tank at a constant volume rate of \(2 \mathrm{~m}^{3} / \mathrm{s}\). Assume that the solution in the tank is well mixed so that the salt concentration in the tank is uniform. Assume also that the salt dissolves completely so that the volume of the mixture remains the same. The salt concentration \(s_{o} \mathrm{~kg} / \mathrm{m}^{3}\) in the outflow is the same as the concentration in the tank. The input is the concentration \(s_{i}(t)\), whose value may change during the process, thus changing the value of \(s_{o}\). Obtain a dynamic model of the concentration \(s_{o}\).

Consider the mixing tank treated in Problem 7.6. Generalize the model to the case where the tank’s volume is \(V \mathrm{~m}^{3}\). For quality control purposes, we want to adjust the output concentration \(s_{o}\) by adjusting the input concentration \(s_{i}\) . How much volume should the tank have so that the change in \(s_{o}\) lags behind the change in \(s_{i}\) by no more than 20 s?

Derive the expression for the fluid capacitance of the cylindrical tank shown in Figure P7.8.

Derive the expression for the capacitance of the container shown in Figure P7.9.

Consider the cylindrical tank shown in Figure P7.8. Derive the dynamic model of the height h, assuming that the input mass flow rate is \(q_{m}(t)\).

Consider the tank shown in Figure P7.9. Derive the dynamic model of the height h, assuming that the input mass flow rate is \(q_{m}(t)\).

Air flows in a certain cylindrical pipe 1 m long with an inside diameter of 1 mm. The pressure difference between the ends of the pipe is 0.1 atm. Compute the laminar resistance, the Reynolds number, the entrance length, and the mass flow rate. Comment on the accuracy of the resistance calculation. For air use \(\mu=1.58 \times 10^{-5} \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}^{2}\) and \(\rho=1.2885 \mathrm{~kg} / \mathrm{m}^{3}\).

Derive the expression for the linearized resistance due to orifice flow near a reference height \(h_{r}\).

Consider the cylindrical container treated in Example 7.4.3. Suppose the outlet flow is turbulent. Derive the dynamic model of the system (a) in terms of the gage pressure p at the bottom of the tank and (b) in terms of the height h.

A certain tank has a bottom area \(A=20 \mathrm{~m}^{2}\). The liquid level in the tank is initially 5 m. When the outlet is opened, it takes 200 s to empty by 98%. a. Estimate the value of the linear resistance R. b. Find the steady-state height if the inflow is \(q=3 \mathrm{~m}^{3} / \mathrm{s}\)

A certain tank has a circular bottom area \(A=20 \ \mathrm{ft}^{2}\). It is drained by a pipe whose linear resistance is \(R=150 \mathrm{~m}^{-1} \mathrm{sec}^{-1}\). The tank contains water whose mass density is \(1.94 \ \mathrm{slug} / \mathrm{ft}^{3}\). a. Estimate how long it will take for the tank to empty if the water height is initially 30 ft. b. Suppose we dump water into the tank at a rate of \(0.1 \ \mathrm{ft}^{3} / \mathrm{sec}\). If the tank is initially empty and the outlet pipe remains open, find the steady-state height and the time to reach one-third that height, and estimate how long it will take to reach the steady-state height.

The water inflow rate to a certain tank was kept constant until the water height above the orifice outlet reached a constant level. The inflow rate was then measured, and the process repeated for a larger inflow rate. The data are given in the table. Find the effective area \(C_{d} A_{o}\) for the tank’s outlet orifice.

In the system shown in Figure P7.18, a component such as a valve has been inserted between the two lengths of pipe. Assume that turbulent flow exists throughout the system. Use the resistance relation 7.3.7. (a) Find the total turbulent resistance. (b) Develop a model for the behavior of the liquid height h, with the mass flow rate \(q_{m i}\) as the input.

The cylindrical tank shown in Figure 7.4.3 has a circular bottom area A. The mass inflow rate from the flow source is \(\hat{q}_{m i}(t)\), a given function of time. The flow through the outlet is turbulent, and the outlet discharges to atmospheric pressure \(p_{a}\). Develop a model of the liquid height h.

In the liquid level system shown in Figure P7.20, the resistances \(R_{1}\) and \(R_{2}\) are linear, and the input is the pressure source \(p_{s}\). Obtain the differential equation model for the height h, assuming that \(\hat{h}>D\).

The water height in a certain tank was measured at several times with no inflow applied. See Figure 7.4.3. The resistance R is a linearized resistance. The data are given in the table. The tank’s bottom area is \(A=6 \ \mathrm{ft}^{2}\). a. Estimate the resistance R. b. Suppose the initial height is known to be exactly 20.2 ft. How does this change the results of part (a)?

Derive the model for the system shown in Figure P7.22. The flow rate \(\hat{q}_{m i}\) is a mass flow rate and the resistances are linearized.

(a) Develop a model of the two liquid heights in the system shown in Figure P7.23. The inflow rate \(q_{m i}(t)\) is a mass flow rate. (b) Using the values \(R_{1}=R, R_{2}=3 R, A_{1}=A\), and \(A_{2}=4 A\), find the transfer function \(H_{2}(s) / Q_{m i}(s)\).

Consider Example 7.4.6. Suppose that \(R_{1}=R, R_{2}=3 R, A_{1}=A\), and \(A_{2}=2 A\). Find the transfer function \(H_{1}(s) / Q_{m i}(s)\) and the characteristic roots.

Design a piston-type damper using an oil with a viscosity at \(20^{\circ} \mathrm{C}\) of \(\mu=0.9 \mathrm{~kg} /(\mathrm{m} \cdot \mathrm{s})\). The desired damping coefficient is \(2000 \mathrm{~N} \cdot \mathrm{s} / \mathrm{m}\). See Figure 7.4.4.

For the damper shown in Figure 7.4.7, assume that the flow through the hole is turbulent, and neglect the term \(m \ddot{y}\). Develop a model of the relation between the force f and \(\dot{x}\), the relative velocity between the piston and the cylinder.

An electric motor is sometimes used to move the spool valve of a hydraulic motor. In Figure P7.27 the force f is due to an electric motor acting through a rack-and-pinion gear. Develop a model of the system with the load displacement y as the output and the force f as the input. Consider two cases: (a) \(m_{1}=0\) and (b) \(m_{1} \neq 0\).

In Figure P7.28 the piston of area A is connected to the axle of the cylinder of radius R, mass m, and inertia I about its center. Develop a dynamic model of the axle’s translation x, with the pressures \(p_{1}\) and \(p_{2}\) as the inputs.

Figure P7.29 shows a pendulum driven by a hydraulic piston. Assuming small angles \(\theta\) and a concentrated mass m a distance \(L_{1}\) from the pivot, derive the equation of motion with the pressures \(p_{1}\) and \(p_{2}\) as inputs.

Figure P7.30 shows an example of a hydraulic accumulator, which is a device for reducing pressure fluctuations in a hydraulic line or pipe. The fluid density is \(\rho\), the plate mass is m, and the plate area is A. Develop a dynamic model of the pressure p with the pressures \(p_{1}\) and \(p_{2}\) as the given inputs. Assume that \(m \ddot{x}\) of the plate is small, and that the hydrostatic pressure \(\rho g h\) is small.

Design a hydraulic accumulator of the type shown in Figure P7.30. The liquid volume in the accumulator should increase by \(30 \text { in. }^{3}\) when the pressure p increases by \(1.5 \ \mathrm{lb} / \mathrm{in} .^{2}\). Determine suitable values for the plate area A and the spring constant k.

Consider the liquid-level system treated in Example 7.4.10 and shown in Figure 7.4.12. The pump curve and the line for the steady-state flow through both valves are shown in Figure P7.32. It is known that the bottom area of the tank is \(2 \mathrm{~m}^{2}\) and the outlet resistance is \(R_{2}=400 \ 1 /(\mathrm{m} \cdot \mathrm{s})\). (a) Compute the pump resistance \(_{1}\) and the steady-state height. (b) Derive a linearized dynamic model of the height deviation \(\delta h\) in the tank.

Consider the V-shaped container treated in Example 7.2.2, whose cross section is shown in Figure P7.33. The outlet resistance is linear. Derive the dynamic model of the height h.

Consider the V-shaped container treated in Example 7.2.2, whose cross section is shown in Figure P7.34. The outlet is an orifice of area \(A_{o}\) and discharge coefficient \(C_{d}\). Derive the dynamic model of the height h.

Consider the cylindrical container treated in Problem 7.8. In Figure P7.35 the tank is shown with a valve outlet at the bottom of the tank. Assume that the flow through the valve is turbulent with a resistance R. Derive the dynamic model of the height h.

A certain tank contains water whose mass density is \(1.94 \ \mathrm{slug} / \mathrm{ft}^{3}\). The tank’s circular bottom area is \(A=100 \ \mathrm{ft}^{2}\). It is drained by an orifice in the bottom. The effective cross-sectional area of the orifice is \(C_{d} A_{o}=0.5 \ \mathrm{ft}^{2}\). A pipe dumps water into the tank at the volume flow rate \(q_{v}\). a. Derive the model for the tank’s height h with the input \(q_{v}\). b. Compute the steady-state height if the input flow rate is \(q_{v}=5 \ \mathrm{ft}^{3} / \mathrm{sec}\). c. Estimate the tank’s time constant when the height is near the steady-state height.

(a) Derive the expression for the fluid capacitance of the conical tank shown in Figure P7.37. The cone angle \(\theta\) is a constant and should appear in your answer as a parameter. (b) Derive the dynamic model of the liquid height h. The mass inflow rate is \(q_{m i}(t)\). The resistance R is linear.

(a) Determine the capacitance of a spherical tank of radius R, shown in Figure P7.38. (b) Obtain a model of the pressure at the bottom of the tank, given the mass flow rate \(q_{m i}\).

Obtain the dynamic model of the liquid height h in a spherical tank of radius R, shown in Figure P7.39. The mass inflow rate through the top opening is \(q_{m i}\) and the orifice resistance is \(R_{o}\).

A rigid container has a volume of \(20 \ \mathrm{ft}^{3}\). The air inside is initially at \(70^{\circ} \mathrm{F}\). Find the pneumatic capacitance of the container for an isothermal process.

Consider the pneumatic system treated in Example 7.5.2. Derive the linearized model for the case where \(p_{i}<p\).

Figure P7.42 shows two rigid tanks whose pneumatic capacitances are \(C_{1}\) and \(C_{2}\). The variables \(\delta p_{i}, \delta p_{1}\), and \(\delta p_{2}\) are small deviations around a reference steady-state pressure \(p_{s s}\). The pneumatic lines have linearized resistances \(R_{1}\) and \(R_{2}\). Assume an isothermal process. Derive a model of the pressures \(\delta p_{1}\) and \(\delta p_{2}\) with \(\delta p_{i}\) as the input.

(a) Compute the thermal capacitance of 250 ml of water, for which \(\rho=1000 \mathrm{~kg} / \mathrm{m}^{3}\) and \(c_{p}=4.18 \times 10^{3} \mathrm{~J} / \mathrm{kg} \cdot{ }^{\circ} \mathrm{C}\). Note that \(1 \mathrm{ml}=10^{-6} \mathrm{~m}^{3}\). (b) How much energy does it take to raise the water temperature from room temperature \(\left(20^{\circ} \mathrm{C}\right)\) to \(99^{\circ} \mathrm{C}\) (just below boiling).

A certain room measures 15 ft by 10 ft by 8 ft. (a) Compute the thermal capacitance of the room air using \(c_{p}=6.012 \times 10^{3} \ \mathrm{ft}-\mathrm{lb} / \text { slug }-{ }^{\circ} \mathrm{F}\) and \(\rho=0.0023 \text { slug } / \mathrm{ft}^{3}\). (b) How much energy is required to raise the air temperature from \(68^{\circ} \mathrm{F}\) to \(72^{\circ} \mathrm{F}\), neglecting heat transfer to the walls, floor, and ceiling?

Liquid initially at \(20^{\circ} \mathrm{C}\) is pumped into a mixing tank at a constant volume flow rate of \(0.5 \mathrm{~m}^{3} / \mathrm{s}\). See Figure 7.6.1. At time t = 0 the temperature of the incoming liquid suddenly is changed to \(80^{\circ} \mathrm{C}\). The tank walls are perfectly insulated. The tank volume is \(12 \mathrm{~m}^{3}\), and the liquid within is well-mixed so that its temperature is uniform throughout, and denoted by T. The liquid’s specific heat and mass density are \(c_{p}\) and \(\rho\). Given that \(T(0)=20^{\circ} \mathrm{C}\), develop and solve a dynamic model for the temperature T as a function of time.

The copper shaft shown in Figure P7.46 consists of two cylinders with the following dimensions: \(L_{1}=10 \mathrm{~mm}, L_{2}=5 \mathrm{~mm}, D_{1}=2 \mathrm{~mm}\), and \(D_{2}=1.5 \mathrm{~mm}\). The shaft is insulated around its circumference so that heat transfer occurs only in the axial direction. (a) Compute the thermal resistance of each section of the shaft and of the total shaft. Use the following value for the conductivity of copper: \(k=400 \mathrm{~W} / \mathrm{m} \cdot{ }^{\circ} \mathrm{C}\). (b) Compute the heat flow rate in the axial direction if the temperature difference across the endpoints of the shaft is \(30^{\circ} \mathrm{C}\).

A certain radiator wall is made of copper with a conductivity \(k=47 \ \mathrm{lb} / \mathrm{sec}-{ }^{\circ} \mathrm{F}\) at \(212^{\circ} \mathrm{F}\). The wall is 3/16 in. thick and has circulating water on one side with a convection coefficient \(h_{1}=85 \ \mathrm{lb} / \mathrm{sec}-\mathrm{ft}-{ }^{\circ} \mathrm{F}\). A fan blows air over the other side, which has a convection coefficient \(h_{2}=15 \ \mathrm{lb} / \mathrm{sec}-\mathrm{ft}-{ }^{\circ} \mathrm{F}\). Find the thermal resistance of the radiator on a square foot basis.

A particular house wall consists of three layers and has a surface area of \(30 \mathrm{~m}^{2}\). The inside layer is 10 mm thick and made of plaster board with a thermal conductivity of \(k=0.2 \mathrm{~W} /\left(\mathrm{m} \cdot{ }^{\circ} \mathrm{C}\right)\). The middle layer is made of fiberglass insulation with \(k=0.04 \mathrm{~W} /\left(\mathrm{m} \cdot{ }^{\circ} \mathrm{C}\right)\). The outside layer is 20 mm thick and made of wood siding with \(k=0.1 \mathrm{~W} /\left(\mathrm{m} \cdot{ }^{\circ} \mathrm{C}\right)\). The inside temperature is \(20^{\circ} \mathrm{C}\), and the convection coefficient for the inside wall surface is \(h_{i}=40 \ \mathrm{W} /\left(\mathrm{m}^{2} \cdot{ }^{\circ} \mathrm{C}\right)\). The convection coefficient for the outside wall surface is \(h_{o}=70 \mathrm{~W} /\left(\mathrm{m}^{2} \cdot{ }^{\circ} \mathrm{C}\right)\). How thick must the insulation layer be so that the heat loss is no greater than 400 W if the outside temperature is \(-20^{\circ} \mathrm{C}\)?

A certain wall section is composed of a 12 in. by 12 in. brick area 4 in. thick. Surrounding the brick is a 36 in. by 36 in. concrete section, which is also 4 in. thick. The thermal conductivity of the brick is \(k=0.086 \ \mathrm{lb} / \mathrm{sec}-{ }^{\circ} \mathrm{F}\). For the concrete, \(k=0.02 \ \mathrm{lb} / \mathrm{sec}-{ }^{\circ} \mathrm{F}\). (a) Determine the thermal resistance of the wall section. (b) Compute the heat flow rate through (1) the concrete, (2) the brick, and (3) the wall section if the temperature difference across the wall is \(40^{\circ} \mathrm{F}\).

Water at \(120^{\circ} \mathrm{F}\) flows in an iron pipe 10 ft long, whose inner and outer radii are 1/2 in. and 3/4 in. The temperature of the surrounding air is \(70^{\circ} \mathrm{F}\). (a) Assuming that the water temperature remains constant along the length of the pipe, compute the heat loss rate from the water to the air in the radial direction, using the following values. For iron, \(k=10.1 \ \mathrm{lb} / \mathrm{sec}-{ }^{\circ} \mathrm{F}\). The convection coefficient at the inner surface between the water and the iron is \(h_{i}=16 \ \mathrm{lb} / \mathrm{sec}-\mathrm{ft}-{ }^{\circ} \mathrm{F}\). The convection coefficient at the outer surface between the air and the iron is \(h_{o}=1.1 \ \mathrm{lb} / \mathrm{sec}-\mathrm{ft}-{ }^{\circ} \mathrm{F}\). (b) Suppose the water is flowing at 0.5 ft/sec. Check the validity of the constant-temperature assumption. For water, \(\rho=1.94 \ \mathrm{slug} / \mathrm{ft}^{3}\) and \(c_{p}=25,000 \ \mathrm{ft}-\mathrm{lb} / \text { slug- }{ }^{\circ} \mathrm{F}\).

Consider the water pipe treated in Example 7.7.4. Suppose now that the water is not flowing. The water is initially at \(120^{\circ} \mathrm{F}\). The copper pipe is 6 ft long, with inner and outer radii of 1/4 in. and 3/8 in. The temperature of the surrounding air is constant at \(70^{\circ} \mathrm{F}\). Neglect heat loss from the ends of the pipe, and use the following values. For copper, \(k=50 \ \mathrm{lb} / \mathrm{sec}-{ }^{\circ} \mathrm{F}\). The convection coefficient at the inner surface between the water and the copper is now different because the water is standing. Use \(h_{i}=6 \ \mathrm{lb} / \mathrm{sec}-\mathrm{ft}-{ }^{\circ} \mathrm{F}\). The convection coefficient at the outer surface between the air and the copper is \(h_{o}=1.1 \ \mathrm{lb} / \mathrm{sec}-\mathrm{ft}^{\circ} \mathrm{F}\). Develop and solve a dynamic model of the water temperature T (t) as a function of time.

A steel tank filled with water has a volume of \(1000 \ \mathrm{ft}^{3}\). Near room temperature, the specific heat for water is \(c=25,000 \ \mathrm{ft}-\mathrm{lb} / \text { slug- }{ }^{\circ} \mathrm{F}\), and its mass density is \(\rho=1.94 \ \mathrm{slug} / \mathrm{ft}^{3}\). a. Compute the thermal capacitance \(C_{1}\) of the water in the tank. b. Denote the total thermal resistance (convective and conductive) of the tank’s steel wall by \(R_{1}\). The temperature of the air surrounding the tank is \(T_{o}\). The tank’s water temperature is \(T_{1}\). Assume that the thermal capacitance of the steel wall is negligible. Derive the differential equation model for the water’s temperature, with \(T_{o}\) as the input.

Consider the tank of water discussed in Problem 7.52. A test was performed in which the surrounding air temperature \(T_{o}\) was held constant at \(70^{\circ} \mathrm{F}\). The tank’s water temperature was heated to \(90^{\circ}\) and then allowed to cool. The following data show the tank’s water temperature as a function of time. Use these data to estimate the value of the thermal resistance \(R_{1}\).

The oven shown in Figure P7.54 has a heating element with appreciable capacitance \(C_{1}\). The other capacitance is that of the oven air \(C_{2}\). The corresponding temperatures are \(T_{1}\) and \(T_{2}\), and the outside temperature is \(T_{o}\). The thermal resistance of the heater-air interface is \(R_{1}\); that of the oven wall is \(R_{2}\). Develop a model for \(T_{1}\) and \(T_{2}\), with input \(q_{i}\), the heat flow rate delivered to the heater mass.

A simplified representation of the temperature dynamics of two adjacent masses is shown in Figure P7.55. The mass with capacitance \(C_{2}\) is perfectly insulated on all sides except one, which has a convective resistance \(R_{2}\). The thermal capacitances of the masses are \(C_{1}\) and \(C_{2}\), and their representative uniform temperatures are \(T_{1}\) and \(T_{2}\). The thermal capacitance of the surroundings is very large and the temperature is \(T_{o}\). (a) Develop a model of the behavior of \(T_{1}\) and \(T_{2}\). (b) Discuss what happens if the thermal capacitance \(C_{2}\) is very small.

A metal sphere 25 mm in diameter was heated to \(95^{\circ} \mathrm{C}\), and then suspended in air at \(22^{\circ} \mathrm{C}\). The mass density of the metal is \(7920 \mathrm{~kg} / \mathrm{m}^{3}\), its specific heat at \(30^{\circ} \mathrm{C}\) is \(c_{p}=500 \mathrm{~J} /\left(\mathrm{kg} \cdot{ }^{\circ} \mathrm{C}\right)\), and its thermal conductivity at \(30^{\circ} \mathrm{C}\) is \(400 \mathrm{~W} /\left(\mathrm{m} \cdot{ }^{\circ} \mathrm{C}\right)\). The following sphere temperature data were measured as the sphere cooled. a. Assume that the sphere’s heat loss rate is due entirely to convection. Estimate the convection coefficient h. b. Compute the Biot number and discuss the accuracy of the lumped-parameter model used in part (a). c. Discuss whether some of the heat loss rate could be due to radiation. Give a numerical reason for your answer.

A copper sphere is to be quenched in an oil bath whose temperature is \(50^{\circ} \mathrm{C}\). The sphere’s radius is 30 mm, and the convection coefficient is \(h=300 \mathrm{~W} /\left(\mathrm{m}^{2} \cdot{ }^{\circ} \mathrm{C}\right)\). Assume the sphere and the oil properties are constant. These properties are given in the following table. The sphere’s initial temperature is \(400^{\circ} \mathrm{C}\). Assume that the volume of the oil bath is large enough so that its temperature does not change when the sphere is immersed in the bath. Obtain the dynamic model of the sphere’s temperature T. How long will it take for T to reach \(130^{\circ} \mathrm{C}\)?

Consider the quenching process discussed in Problem 7.57. Suppose the oil bath volume is \(0.1 \mathrm{~m}^{3}\). Neglect any heat loss to the surroundings and develop a dynamic model of the sphere’s temperature and the bath temperature. How long will it take for the sphere temperature to reach \(130^{\circ} \mathrm{C}\)?

Consider Example 7.7.1. The MATLAB left division operator can be used to solve the set of linear algebraic equations AT = b as follows: T=A\b. Use this method to write a script file to solve for the three steady-state temperatures \(T_{1}, T_{2}\), and \(T_{3}\), given values for the resistances and the temperatures \(T_{i}\) and \(T_{o}\). Use the results of Example 7.7.1 to test your file.

Fluid flows in pipe networks can be analyzed in a manner similar to that used for electric resistance networks. Figure P7.60a shows a network with three pipes, which is analogous to the electrical network shown in part (b) of the figure. The volume flow rates in the pipes are \(q_{1}, q_{2}\), and \(q_{3}\). The pressures at the pipe ends are \(p_{a}, p_{b}\), and \(p_{c}\). The pressure at the junction is \(p_{1}\). a. Assuming that the linear resistance relation applies, we have \(q_{1}=\frac{1}{R_{1}}\left(p_{a}-p_{1}\right)\) Obtain the equations for \(q_{2}\) and \(q_{3}\). b. Note that conservation of mass gives \(q_{1}=q_{2}+q_{3}\). Set up the equations in a matrix form Aq = b suitable for solving for the three flow rates \(q_{1}, q_{2}\), and \(q_{3}\), and the pressure \(p_{1}\), given the values of the pressures \(p_{a}, p_{b}\), and \(p_{c}\), and the values of the resistances \(R_{1}, R_{2}\), and \(R_{3}\). Find the expressions for matrix A and the column vector b. c. Use MATLAB to solve the matrix equations obtained in part (b) for the case: \(p_{a}=30 \ \mathrm{psi}, p_{b}=25 \ \mathrm{psi}\), and \(p_{c}=20 \ \mathrm{psi}\). Use the resistance values \(R_{1}=10,000, R_{2}=R_{3}=14,000 \ 1 /(\mathrm{ft}-\mathrm{sec})\). These values correspond to fuel oil flowing through pipes 2 ft long, with 2 in. and 1.4 in. diameters, respectively. The units of the answers should be \(\mathrm{ft}^{3} / \mathrm{sec}\) for the flow rates, and \(\mathrm{lb} / \mathrm{ft}^{2}\) for pressure.

The equation describing the water height h in a spherical tank with a drain at the bottom is \(\pi\left(2 r h-h^{2}\right) \frac{d h}{d t}=-C_{d} A_{o} \sqrt{2 g h}\) Suppose the tank’s radius is r = 3 m and that the circular drain hole has a radius of 2 cm. Assume that \(C_{d}=0.5\), and that the initial water height is h(0) = 5 m. Use \(g=9.81 \mathrm{~m} / \mathrm{s}^{2}\). a. Use an approximation to estimate how long it takes for the tank to empty. b. Use MATLAB to solve the nonlinear equation and plot the water height as a function of time until h(t) is not quite zero.

The following equation describes a certain dilution process, where y(t) is the concentration of salt in a tank of fresh water to which salt brine is being added. \(\frac{d y}{d t}+\frac{2}{10+2 t} y=4\) Suppose that y(0) = 0. a. Use MATLAB to solve this equation for y(t) and to plot y(t) for \(0 \leq t \leq 10\). b. Check your results by using an approximation that converts the differential equation into one having constant coefficients.

A tank having vertical sides and a bottom area of \(100 \ \mathrm{ft}^{2}\) is used to store water. To fill the tank, water is pumped into the top at the rate given in the following table. Use MATLAB to solve for and plot the water height h(t) for \(0 \leq t \leq 10 \mathrm{~min}\).

A cone-shaped paper drinking cup (like the kind used at water fountains) has a radius R and a height H. If the water height in the cup is h, the water volume is given by \(V=\frac{1}{3} \pi\left(\frac{R}{H}\right)^{2} h^{3}\) Suppose that the cup’s dimensions are R = 1.5 in. and H = 4 in. a. If the flow rate from the fountain into the cup is \(2 \text { in. }^{3} / \mathrm{sec}\), use MATLAB to determine how long will it take to fill the cup to the brim. b. If the flow rate from the fountain into the cup is given by \(2\left(1-e^{-2 t}\right) \text { in. }{ }^{3} / \mathrm{sec}\), use MATLAB to determine how long will it take to fill the cup to the brim.

Refer to Figure 7.10.1. Assume that the resistances obey the linear relation, so that the mass flow \(q_{l}\) through the left-hand resistance is \(q_{l}=\left(p_{l}-p\right) / R_{l}\), with a similar linear relation for the right-hand resistance. a. Create a Simulink subsystem block for this element. b. Use the subsystem block to create a Simulink model of the system discussed in Example 7.4.3 and shown in Figure 7.4.3a. Assume that the mass inflow rate \(q_{m i}\) is a step function. c. Use the Simulink model to obtain plots of \(h_{1}(t)\) and \(h_{2}(t)\) for the following parameter values: \(A_{1}=2 \mathrm{~m}^{2}, A_{2}=5 \mathrm{~m}^{2}, R_{1}=4001 /(\mathrm{m} \cdot \mathrm{s})\), R_{2}=6001 /(\mathrm{m} \cdot \mathrm{s}), \(\rho=1000 \mathrm{~kg} / \mathrm{m}^{3}, q_{m i}=50 \mathrm{~kg} / \mathrm{s}, h_{1}(0)=1.5 \mathrm{~m}\), and \(h_{2}(0)=0.5 \mathrm{~m}\).

Use Simulink to solve Problem 7.61(b).

Use Simulink to solve Problem 7.63.

Use Simulink to solve Problem 7.64. Plot h(t) for both parts (a) and (b).

Refer to Example 7.10.1. Use the simulation with \(q=8 \times 10^{5}\) to compare the energy consumption and the thermostat cycling frequency for the two temperature bands \(\left(69^{\circ}, 71^{\circ}\right)\) and \(\left(68^{\circ}, 72^{\circ}\right)\).

Consider the liquid-level system shown in Figure 7.3.3. Suppose that the height h is controlled by using a relay to switch the flow rate \(q_{m i}\) between the values 0 and 50 kg/s. The flow rate is switched on when the height is less than 4.5 m and is switched off when the height reaches 5.5 m. Create a Simulink model for this application using the values \(A=2 \mathrm{~m}^{2}, R=4001 /(\mathrm{m} \cdot \mathrm{s})\), \(\rho=1000 \mathrm{~kg} / \mathrm{m}^{3}\), and h(0) = 1 m. Obtain a plot of h(t).