7.1 Thermal Block Question
A block of copper is brought out of an oven at an initial temperature of 1000 K, and allowed to cool to the ambient temperature of 300 K. The graph below shows data which might represent this experiment. The copper block has a mass of 0.1 kg. Given that copper has a specific heat of 385
Figure 7.1.1: The block temperature (Tblock) is plotted against time (t). The ambient temperature is given by Tamb.
7.2 The Hot Copper Block 1
This problem considers the thermal dynamics of water and a copper block inside a vacuum bottle.
A diagram for the system is shown below in Figure 7.2.1.
Figure 7.2.1: Copper block and water thermal system
Initially, a 100 gm copper block is heated outside the bottle in an oven to a temperature of 95◦C.
We are not concerned with the heating process. The water in the vacuum bottle has been there for a long time and is equilibrated to the ambient temperature Ta = 25◦C.
At t = 0, the stopper is removed, the copper block is lowered into the bottle, and the stopper is replaced. Now we would like to develop a model which can describe the temperature dynamics as a function of time.
The system parameters are as follows: The copper block has a mass of 100 gm; copper has a specific heat of 0.385 J/gm◦C= 385J/kg◦C. The bottle contains 1 liter of water; water has a specific heat of 4.2 J/gm◦C= 4200J/kg◦C.
The temperature of the block is defined as Tb. The water is assumed uniform in temperature, with a value of Tw.
The heat flow from the block to the water is defined as qbw [W]. The heat flow from the water to ambient is defined as qwa [W]. These flows are modeled as passing through thermal resistances Rbw = 0.1◦C/W and Rwa = 17◦C/W, respectively. Note: Take a look at the 2nd order ther
mal example on the course web page before attempting this problem.
73
(a) Write the governing differential equations for the system in state-space form as dTw
= f1(Tw, Tb, Ta) dt
dTb
= f2(Tw, Tb) dt
where f1 and f2 are linear functions.
(b) Convert this representation to a 2nd order differential equation in Tw(t). You may assume that the ambient temperature is constant, and thus dTdt a = 0.
(c) Solve for the response Tw(t) for t ≥ 0. What are the system natural frequencies? Plot the system poles on an s-plane plot.
(d) Use Matlab to plot Tw(t), and its two constituent modes, over an interesting time range.
You may need to use two plots with different time scales to capture the dynamics. Explain qualitatively why the plot looks as it does. Can you show us the effect of the system time constants?
(e) Suppose the block were initially at only 60◦C. Make a sketch on your time response plot of the resulting Tw(t). Explain your reasoning. (Hint: If you exploit linearity, this should require no new calculations.)
7.3 The Hot Copper Block 2
This problem is a continuation of Problem 7.2. Reconsider the system of Problem 7.2, but with an electrical heater added to the inside of the copper block. The heater provides an input power qin = 4 W. This new system is shown in Figure 7.3.1. The system starts at t = 0 in a rest state with Tb = Tw = Ta = 25◦C. At t = 0, the heater is turned on.
qin
Figure 7.3.1: Copper block and water thermal system with heater
(a) Write the governing differential equation in terms of water temperature Tw.
(b) Solve the differential equation for Tw(t), with the numerical values given in Problem 7.2.
(c) Use MATLAB to plot Tw(t) over both a short and long time interval to show the interesting parts of the graphs. What is the steady-state value of Tw? What is the 10% to 90% rise time tr? How long does the response take to settle within 1% of the final value?
75
7.4 Transistor on heat sink
Figure 7.4.1 is a schematic cross-section of a power transistor mounted on a heat sink. The tran
sistor itself is fabricated on a thin piece of silicon, perhaps 5mm x 5mm x 0.5mm. This device is mounted within a package (case) which provides mechanical protection as well as electrical and thermal connections to the outside world. The case is mounted on a finned heat sink in order to transfer heat dissipation in the transistor into the ambient air.
Figure 7.4.1: Transistor Schematic
The case of most transistors needs to be electrically insulated from the heat sink. This is ac
complished with the indicated thermal washer, which has the conflicting requirements of thermal conduction and electrical insulation.
We define the device temperature as Td, the case temperature as Tc, and the heat sink temperature as Ts. The ambient temperature is Ta = 25◦C. The thermal resistance between the device and case is Rdc, between the case and heat sink is Rcs, and between the heat sink and air is Rsa.
The transistor is used as an electrical switch or amplifier. In the process electrical power is converted to heat. The heat flow needs to be managed without allowing the device to become too hot (generally the device temperature should stay below 100◦C).
(a) Assume Rdc = 0.1◦C/W, Rcs = 0.2◦C/W, and Rsa = 0.5◦C/W. Further assume that in steady-state the transistor is dissipating 50W. What are the steady-state values of Td, Tc, and Ts?
(b) Now assume the heat sink is constructed with 500gm of aluminum, which has a specific heat of 0.90 J/g◦K. The system is operating in the steady-state condition above when, at t = 0 the
50W. Write expressions for Td(t), Tc(t), and Ts(t) for t ≥ 0, and make plots of these functions.
Assume that the device and case have zero thermal capacitance. What is the peak device temperature in this linear model. Will it likely survive the transient? Would a bigger heat sink help?
77