Question 10.7: KNOWN: Simple expression to account for the effect of pressu......

ANALYSIS: The simple expression by Jakob [51] accounting for pressure effects is

h = C \left(\Delta T _{ e }\right)^{ n }\left( p / p _{ a }\right)^{0.4}                                                                             (1)

where \mathrm p~\text{and}~\mathrm p_{\mathrm a} are the system and standard atmospheric pressures. For a horizontal plate, C = 5.56 and n = 3 for the range 15 < \mathrm q_{\mathrm s}^{\prime\prime} < 235 kW/m². For Δ \mathrm T_{\mathrm e} = 10°C,

p = 2 bar      h =5.56(10)^3(2  bar / 1.0133  bar )^{0.4}=7,298  W / m ^2 \cdot K , \quad q _{ s }^{\prime \prime}=73  kW / m ^2

p = 5 bar      h =5.56(10)^3(5  bar / 1.0133  bar )^{0.4}=10,529 W / m ^2 \cdot K , \quad q _{ s }^{\prime \prime}=105  kW / m ^2

where \mathrm q_{\mathrm s}^{\prime\prime} = \mathrm h\Delta\mathrm T_{\mathrm e}. The Rohsenow correlation, Eq. 10.5, with \mathrm C_{\mathrm{s,f}} = 0.013 and n = 1, is of the form

q _{ s }^{\prime \prime}=\mu_{\ell} h _{ fg }\left[\frac{ g \left(\rho_{\ell}~-~\rho_{ v }\right)}{\sigma}\right]^{1 / 2}\left[\frac{ c _{ p , \ell} \Delta T _{ e }}{ C _{ s , f } h _{ fg } \operatorname{Pr}_{\ell}^{ n }}\right]^3.                                         (2)

p = 2 bar:      q _{ s }^{\prime \prime}=230.7 \times 10^{-6} \frac{ N \cdot s }{ m ^2} \times 2203 \times 10^3 \frac{ J }{ kg }\left[\frac{9.8 \frac{ m }{ s ^2}(942.7  ~-  ~1.1082) \frac{ kg }{ m ^3}}{54.97~ \times ~10^{-3}  N / m }\right]^{1 / 2} \times\left[\frac{4244.3  J / kg \cdot K~ \times~ 10  K }{0.013 ~\times~ 2203 ~\times~ 10^3 \frac{ J }{ kg }~ \times ~1.43^1}\right]^3

                       q _{ s }^{\prime \prime}=232  kW / m ^2

p = 4 bar:      q _{ s }^{\prime \prime}=439  kW / m ^2.

COMMENTS: For ease of comparison, the results with \mathrm p_{\mathrm a} = 1.0133 bar are:

Note that the range of \mathrm q_{\mathrm s}^{\prime\prime} is within the limits of the Simple correlation. The comparison is poor and therefore the correlation is not to be recommended. By manipulation of the Rohsenow results, find that the (\mathrm p/\mathrm p_{\mathrm o})^{\mathrm m} dependence provides m ≈ 0.75, compared to the exponent of 0.4 in the Simple correlation.