Known The electronic components of a computer are cooled by air flowing through a fan mounted at the inlet of the electronics enclosure. Conditions are specified for the air at the inlet and exit. The power required by the electronics and the fan is also specified.
Find Determine the smallest fan area for which the specified limits are met.
Schematic and Given Data:
Engineering Model
1. The control volume shown on the accompanying figure is at steady state.
2. Heat transfer from the outer surface of the electronics enclosure to the surroundings is negligible. Thus, \dot{Q}_{ cv }=0.
1 3. Changes in kinetic and potential energies can be ignored.
2 4. Air is modeled as an ideal gas with c_{p}=1.005 kJ / kg \cdot K.
Analysis The inlet area A _{1} can be determined from the mass flow rate \dot{m} and Eq. 4.4b, which can be rearranged to read
\dot{m}=\frac{ AV }{v} (one-dimensional flow) (4.4b)
A _{1}=\frac{\dot{m} v_{1}}{ V _{1}} (a)
The mass flow rate can be evaluated, in turn, from the steadystate energy rate balance, Eq. 4.20a.
0=\dot{Q}_{ cv }-\dot{W}_{ cv }+\dot{m}\left[\left(h_{1}-h_{2}\right)+\frac{\left( V _{1}^{2}- V _{2}^{2}\right)}{2}+g\left(z_{1}-z_{2}\right)\right] (4.20a)
0=\underline{\dot{Q}_{ cv }}-\dot{W}_{ cv }+\dot{m}\left[\left(h_{1}-h_{2}\right)+\underline{\left(\frac{ V _{1}^{2}- V _{2}^{2}}{2}\right)}+\underline{g\left(z_{1}-z_{2}\right)}\right]
The underlined terms drop out by assumptions 2 and 3, leaving
0=-\dot{W}_{ cv }+\dot{m}\left(h_{1}-h_{2}\right)
where \dot{W}_{ cv } accounts for the total electric power provided to the electronic components and the fan: \dot{W}_{ cv }=(-80 W )+(-18 W )=-98 W. Solving for \dot{m}, and using assumption 4 with Eq. 3.51 to evaluate \left(h_{1}-h_{2}\right)
h\left(T_{2}\right)-h\left(T_{1}\right)=c_{p}\left(T_{2}-T_{1}\right) (3.51)
\dot{m}=\frac{\left(-\dot{W}_{ cv }\right)}{c_{p}\left(T_{2}-T_{1}\right)}
Introducing this into the expression for A _{1}, Eq. (a), and using the ideal gas model to evaluate the specific volume v_{1}
A _{1}=\frac{1}{ V _{1}}\left[\frac{\left(-\dot{W}_{ cv }\right)}{c_{p}\left(T_{2}-T_{1}\right)}\right]\left(\frac{R T_{1}}{p_{1}}\right)
From this expression we see that A _{1} increases when V _{1} and/or T _{2} decrease. Accordingly, since V _{1} \leq 1.3 m / s \text { and } T_{2} \leq 305 K (32°C), the inlet area must satisfy
A _{1} \geq \frac{1}{1.3 m / s }\left[\frac{98 W }{\left(1.005 \frac{ kJ }{ kg \cdot K }\right)(305-293) K }\left|\frac{1 kJ }{10^{3} J }\right| \left|\frac{1 J / s }{1 W }\right|\right]
\times\left(\frac{\left(\frac{8314 N \cdot m }{28.97 kg \cdot K }\right) 293 K }{1.01325 \times 10^{5} N / m ^{2}}\right)\left|\frac{10^{4} cm ^{2}}{1 m ^{2}}\right|
\geq 52 cm ^{2}
For the specified conditions, the smallest fan area is 52 cm ^{2}.
1 Cooling air typically enters and exits electronic enclosures at low velocities, and thus kinetic energy effects are insignificant.
2 The applicability of the ideal gas model can be checked by reference to the generalized compressibility chart. Since the temperature of the air increases by no more than 12°C, the specific heat c_{p} is nearly constant (Table A-20).
Skills Developed
Ability to…
• apply the steady-state energy rate balance to a control volume.
• apply the mass flow rate expression, Eq. 4.4b.
• develop an engineering model.
• retrieve property data of air modeled as an ideal gas.
Quick Quiz
If heat transfer occurs at a rate of 11 W from the outer surface of the computer case to the surroundings, determine the smallest fan inlet area for which the limits on entering air velocity and exit air temperature are met if the total power input remains at 98 W. Ans. 46 cm ^{2}
