Temperature Dynamics of a House with a Heater The room shown in Figure 7.30 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

Question

Temperature Dynamics of a House with a Heater

The room shown in Figure 7.30 has a heater with heat flow rate input of [latex]q_{0}[/latex]. The thermal capacitances of the heater and the room air are [latex]C_{1}[/latex] and [latex]C_{2}[/latex], respectively. The thermal resistances of the heater-air interface and the room wall-ambient air interface are [latex]R_{1}[/latex] and [latex]R_{2}[/latex], respectively. The temperatures of the heater and the room air are [latex]T_{1}[/latex] and [latex]T_{2}[/latex], respectively. The temperature outside the room is [latex]T_{0}[/latex], which is assumed to be constant.

a. Derive the differential equations relating the temperatures [latex]T_{1}, T_{2}[/latex], the input [latex]q_{0}[/latex], and the outside temperature [latex]T_{0}[/latex].

b. Using the differential equations obtained in Part (a), determine the state-space form of the system. Assume the temperatures [latex]T_{1}[/latex] and [latex]T_{2}[/latex] as the outputs.

Accepted Answer

a. Applying the law of conservation of energy to the heater, we have

[latex]\frac{\mathrm{d} U}{\mathrm{~d} t}=q_{\mathrm{hi}}-q_{\mathrm{h} o^{\prime}}[/latex]

where

[latex] \begin{aligned} \frac{\mathrm{d} U}{\mathrm{~d} t} & =C_{1} \frac{\mathrm{d} T_{1}}{\mathrm{~d} t}, \ q_{\mathrm{hi}} & =q_{0}, \ q_{\mathrm{ho}} & =\frac{T_{1}-T_{2}}{R_{1}} . \end{aligned} [/latex]

Substituting these expressions gives

[latex]C_{1} \frac{\mathrm{d} T_{1}}{\mathrm{~d} t}=q_{0}-\frac{T_{1}-T_{2}}{R_{1}},[/latex]

which can be rearranged into

[latex]C_{1} \frac{\mathrm{d} T_{1}}{\mathrm{~d} t}+\frac{1}{R_{1}} T_{1}-\frac{1}{R_{1}} T_{2}=q_{0}.[/latex]

Applying the law of conservation of energy to the room air, we have

[latex]\frac{\mathrm{d} U}{\mathrm{~d} t}=q_{\mathrm{hi}}-q_{\mathrm{ho^{\prime}}}[/latex]

where

[latex] \begin{aligned} \frac{\mathrm{d} U}{\mathrm{~d} t} & =C_{2} \frac{\mathrm{d} T_{2}}{\mathrm{~d} t}, \ q_{\mathrm{hi}} & =\frac{T_{1}-T_{2}}{R_{1}}, \ q_{\mathrm{ho}} & =\frac{T_{2}-T_{0}}{R_{2}} . \end{aligned} [/latex]

Substituting these expressions gives

[latex]C_{2} \frac{\mathrm{d} T_{2}}{\mathrm{~d} t}=\frac{T_{1}-T_{2}}{R_{1}}-\frac{T_{2}-T_{0}}{R_{2}},[/latex]

which can be rearranged into

[latex]C_{2} \frac{\mathrm{d} T_{2}}{\mathrm{~d} t}-\frac{1}{R_{1}} T_{1}+\left(\frac{1}{R_{1}}+\frac{1}{R_{2}} ight) T_{2}=\frac{1}{R_{2}} T_{0} .[/latex]

The system of differential equations can be written in second-order matrix form as

[latex] \left[\begin{array}{cc} C_{1} & 0 \ 0 & C_{2} \end{array} ight]\left[\begin{array}{c} \frac{\mathrm{d} T_{1}}{\mathrm{~d} t} \ \frac{\mathrm{d} T_{2}}{\mathrm{~d} t} \end{array} ight\}+\left[\begin{array}{cc} \frac{1}{R_{1}} & -\frac{1}{R_{1}} \ -\frac{1}{R_{1}} & \left(\frac{1}{R_{1}}+\frac{1}{R_{2}} ight) \end{array} ight]\left\{\begin{array}{c} T_{1} \ T_{2} \end{array} ight\}=\left\{\begin{array}{c} q_{0} \ \frac{1}{R_{2}} T_{0} \end{array} ight\} . [/latex]

b. To represent a thermal system in the state-space form, the temperature of each thermal capacitance is often chosen as a state variable. As specified, the state, the input, and the output are

[latex] \mathbf{x}=\left\{\begin{array}{l} x_{1} \ x_{2} \end{array} ight\}=\left\{\begin{array}{l} T_{1} \ T_{2} \end{array} ight\}, \quad \mathbf{u}=\left\{\begin{array}{l} q_{0} \ T_{0} \end{array} ight\}, \quad \mathbf{y}=\left\{\begin{array}{l} T_{1} \ T_{2} \end{array} ight\} . [/latex]

The state-variable equations are

[latex] \begin{aligned} & \dot{x}_{1}=\frac{\mathrm{d} T_{1}}{\mathrm{~d} t}=-\frac{1}{R_{1} C_{1}} T_{1}+\frac{1}{R_{1} C_{1}} T_{2}+\frac{1}{C_{1}} q_{0} \ & \dot{x}_{2}=\frac{\mathrm{d} T_{2}}{\mathrm{~d} t}=\frac{1}{R_{1} C_{2}} T_{1}-\left(\frac{1}{R_{1} C_{2}}+\frac{1}{R_{2} C_{2}} ight) T_{2}+\frac{1}{R_{2} C_{2}} T_{0}, \end{aligned} [/latex]

or in matrix form

[latex] \left\{\begin{array}{l} \dot{x}_{1} \ \dot{x}_{2} \end{array} ight\}=\left[\begin{array}{cc} -\frac{1}{R_{1} C_{1}} & \frac{1}{R_{1} C_{1}} \ \frac{1}{R_{1} C_{2}} & -\left(\frac{1}{R_{1} C_{2}}+\frac{1}{R_{2} C_{2}} ight) \end{array} ight]\left\{\begin{array}{l} x_{1} \ x_{2} \end{array} ight\}+\left[\begin{array}{cc} \frac{1}{C_{1}} & 0 \ 0 & \frac{1}{R_{2} C_{2}} \end{array} ight]\left\{\begin{array}{l} u_{1} \ u_{2} \end{array} ight\} . [/latex]

The output equation is

[latex] \left\{\begin{array}{l} y_{1} \ y_{2} \end{array} ight\}=\left[\begin{array}{ll} 1 & 0 \ 0 & 1 \end{array} ight]\left\{\begin{array}{l} x_{1} \ x_{2} \end{array} ight\}+\left[\begin{array}{ll} 0 & 0 \ 0 & 0 \end{array} ight]\left\{\begin{array}{l} u_{1} \ u_{2} \end{array} ight\} . [/latex]

Question 7.8: Temperature Dynamics of a House with a Heater The room shown...

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