A vertical double wall 2.5m high 1m wide has an air gap 2.5 cm thick. The internal wall faces spanning the air gap are at 305K and 295 K, respectively. Determine the heat loss across the air gap.

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We make use of ElShirbiny et al. correlation to solve this problem. The given data, in the usual notation is

s = 2.5 cm = 0.025m , H = 2.5m , [latex]T_{h}=305 K[/latex] , [latex]T_{c}=295 K[/latex]

The aspect ratio of the vertical enclosure is

[latex]\frac{H}{s}=\frac{2.5}{0.025}=100[/latex]

Mean temperature of trapped air is [latex]T_{m}=\frac{T_{h}+T_{c}}{2}=\frac{305+205}{2}= 300 K[/latex] . Air properties at 300K are

[latex]v_{m}=15.89 imes 10^{-6} m ^{2} / s[/latex] , [latex]\operatorname{Pr}_{m}=0.7[/latex] , [latex]k_{m}=0.0263 W / m ^{\circ} C[/latex] , [latex]\beta_{m}=\frac{1}{300}=0.0033 K ^{-1}[/latex]

The Rayleigh number is calculated as

[latex]R a_{s}=\frac{9.81 imes 0.0033(305-295) 0.025^{3}}{\left(15.89 imes 10^{-6} ight)^{2}} imes 0.7=14165[/latex]

We have to find the biggest of the three alternate expressions given by Eq. 16.66

[latex]\overline{N u}_{s}=0.06 R a_{s}^{\frac{1}{3}} ext { or } \overline{N u}_{s}=\left[1+\left\{\frac{0.104 R a_{s}^{0.293}}{1+\left(\frac{6310}{R a_{s}} ight)^{1.36}} ight\}^{3} ight]^{\frac{1}{3}} ext { or } \overline{N u}_{s}=0.242\left[\frac{R a_{s}}{\left(\frac{H}{s} ight)} ight]^{0.272}[/latex] (16.66)

(a) [latex]\overline{N u}_{s}=0.0605 imes 14165^{\frac{1}{3}}=1.464[/latex]

(b) [latex]\overline{N u}_{s}==\left[1+\left\{\frac{0.104 imes 14165^{0.293}}{1+\left(\frac{6310}{14165} ight)^{1} .36} ight\}^{3} ight]^{\frac{1}{3}}=1.461[/latex]

(c) [latex]\overline{N u}_{s}=0.242\left[\frac{14165}{100} ight]^{0.272}=0.931[/latex]

Choosing the biggest of these, we have [latex]\overline{N u_{s}}=1.464[/latex]. Hence, the heat transfer coefficient is

[latex]\bar{h} =\frac{1.464 imes 0.0263}{0.025}=1.54[/latex]

The heat transfer across the air layer is

[latex]Q=\bar{h} H W\left(T_{h}-T_{c} ight)=1.54 imes 2.5 imes 1(305-295)=38.5 W[/latex]

Question 16.9: A vertical double wall 2.5m high 1m wide has an air gap 2.5 ...

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