Category Archives: Accounting

erive the differential equations in terms of the liquid heights h1 and h2.

 

Figure 7.18 shows a liquid-level system in which two tanks have cross-sectional areas A1 and A2, respectively. The volume flow rate into tank 1 is qi. A pump is connected to the bottom of tank 1, and the pressure of the fluid increases by Δp when crossing the pump. Tank 2 is located higher than tank 1, and the vertical distance between the two tanks is H. The liquid is pumped from tank 1 to tank 2 through a valve of linear resistance R1 and leaves tank 2 through a valve of linear resistance R2. The density ρ of the liquid is constant. Derive the differential equations in terms of the liquid heights h1 and h2. Write the equations in second-order matrix form.

Derive the differential equation relating the liquid height h and the volume flow rate qi at the inlet.

 

Consider the single-tank liquid-level system shown in Figure 7.19, where the volume flow rate into the tank through a pipe is qi. The liquid leaves the tank through an orifice of area Ao. Denote Cd as the discharge coefficient, which is the ratio of the actual mass flow rate to the theoretical one, and lies in the range of 0 <>Cd <>1 because of friction effects. Derive the differential equation relating the liquid height h and the volume flow rate qi at the inlet. The tank’s cross-sectional area is constant. The density ρ of the liquid is constant.

Derive the differential equations in terms of the liquid heights h1 and h2.

Figure 7.20 shows a hydraulic system of two interconnected tanks that have the same cross-sectional area of A. A pump is connected to tank 1. Assume that the relationship between the voltage applied to the pump and the mass flow rate into tank 1 is linear; that is, qmi = kpva, where kp is called the pump constant and can be obtained by experimental measurements. Tank 1 is connected to tank 2, which is connected to a reservoir. The liquid leaves each tank through an outlet of area Ao at the bottom. Derive the differential equations in terms of the liquid heights h1 and h2.

Determine the heat flow rate through the wall.

Consider heat transfer through an insulated frame wall of a house. The thermal conductivity of the wall is 0.055 W/(m・°C). The wall is 0.15 m thick and has an area of 15 m2. The inside air temperature is 20°C and the heat transfer coefficient for convection between the wall and the inside air is 2.6 W/(m・°C). On the outside of the wall, the heat transfer coefficient for convection between the wall and the outside air is 10.4 W/(m・°C) and the outside air temperature is –10°C. Determine the heat flow rate through the wall.

Determine the heat flow rate through the double-pane window.

 

Consider heat transfer through a double-pane window as shown in Figure 7.31a. Two layers of glass with thermal conductivity k1 are separated by a layer of stagnant air with thermal conductivity k2. The inner surface of the window is at temperature T1 and exposed to room air with heat transfer coefficient h1. The outer surface of the wall is at temperature T2 and exposed to air with heat transfer coefficient h2. Assume that k1 = 0.95 W/(m・°C),

k2 = 0.0285 W/(m・°C), h1 = h2 = 10 W/(m2・°C), T1 = 20°C, and T2 = 35°C. The thickness of each glass layer is 4 mm, the thickness of the air layer is 8 mm, and the cross-sectional area

of the window is 1.5 m2.

a. Determine the heat flow rate through the double-pane window.

b. Determine the temperature distribution through the double-pane window.

c. Repeat Parts (a) and (b)….

Determine if the junction’s temperature can be considered uniform.

 

The junction of a thermocouple can be approximated as a sphere with a diameter of 1 mm. As shown in Figure 7.32, the thermocouple is used to measure the temperature of a gas stream. For the junction, the density is ρ = 8500 kg/m3, the specific heat capacity is

c = 320 J/(kg・°C), and the thermal conductivity is k = 40 W/(m・°C). The temperature of the gas Tf is 120°C and the initial temperature of the sphere T0 is 25°C. The heat transfer coefficient between the gas and the junction is h = 70 W/(m2・°C).

a. Determine if the junction’s temperature can be considered uniform.

b. Derive the differential equation relating the junction’s temperature T(t) and the gas’s temperature Tf.

c. Using the differential equation obtained in Part (b), construct a Simulink block diagram to find out….

Derive the differential equation relating the milk temperature T(t) and the water temperature.

 

Figure 7.33 shows a thin-walled glass of milk, which is taken out of the refrigerator at a uniform temperature of 3°C and is placed in a large pan filled with hot water at 60°C. Assume that the assumption of the lumped system analysis is applicable because the milk is stirred constantly, so that its temperature is uniform at all times. The glass container is cylindrical in shape, with a radius of 3 cm and a height of 6 cm. The estimated parameters of the milk are density ρ = 1035 kg/m3, specific heat capacity c = 3980 J/(kg・°C), and thermal conductivity k = 0.56 W/(m・°C). The heat transfer coefficient between the water and the glass is h = 250 W/(m2・°C).

a. Derive the differential equation relating the milk temperature T(t) and the water temperature.

….

Derive the differential equations relating the temperatures T1, T2, the input q0, and the outside temperature T0.

 

The room shown in Figure 7.34 has a heater with heat flow rate input of q0. The thermal capacitances of the heater and the room air are C1 and C2, respectively. The thermal resistances of the heater–air interface and the room wall–ambient air interface are R1 and R2, respectively. The temperatures of the heater and the room air are T1 and T2, respectively. The temperature outside the room is T0, which is assumed to be constant.

a. Derive the differential equations relating the temperatures T1, T2, the input q0, and the outside temperature T0.

b. Using the differential equations obtained in Part (a), determine the state-space form of the system. Assume the temperatures T1 and T2 as the outputs.

Develop a mathematical model of the pressure p in the container.

 

Dry air at a constant temperature of 25°C passes through a valve out of a rigid cubic container of 1.5 m on each side (Figure 7.42). The pressure po at the outlet of the valve is constant, and it is less than p. The valve resistance is approximately linear, and

R = 1000 Pa・s/kg. Assume that the process is isothermal.

a. Develop a mathematical model of the pressure p in the container.

b. Construct a Simulink block diagram to find the output p(t) of the pneumatic system if the pressure inside the container is initially 2 atm and the pressure at the outlet is 1 atm.

Derive the differential equations in terms of the liquid heights h1 and h2

 

Figure 7.43 shows a liquid-level system in which two tanks have hydraulic capacitances

C1 and C2, respectively. The volume flow rate into tank 1 is qi. The liquid flows from tank 1 to tank 2 through a valve of linear resistance R1 and leaves tank 2 through a valve of linear resistance R2. The density ρ of the liquid is constant.

a. Derive the differential equations in terms of the liquid heights h1 and h2. Write the equations in matrix form.

b. Assume that the volume flow rate qi is the input and the liquid heights h1 and h2 are the outputs. Determine the state-space form of the system.

c. Construct a Simulink block diagram to find the outputs h1(t) and h2(t) of the liquid-level system. Assume that ρ = 1000 kg/m3, g = 9.81 m/s2, C1 = 0.15 kg・m2/N, C2 = 0.25 kg・m2/N,

R1 = R2 = 100 N・s/(kg・m2), and initial liquid….