Heat transfer predictions for forced convective condensation

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Heat transfer characteristics for convective condensation in a tube depend strongly on the flow regimes and the different structures of their liquid-vapor interfaces. Heat transfer in stratified and annular regimes has been extensively studied, and different correlations based on theoretical analysis and experimental data have been proposed.

Gravitational effects can be neglected for the high mass flux region, ( ${\dot{m}}''>400\text{ kg/}{{\text{m}}^{\text{2}}}\text{-s}$ ), so that annular flow with convective condensation in a horizontal tube can be analyzed with the method applied to the vertical tube.For practical applications, the following empirical correlation, proposed by Shah (1979) from a fit of 474 experimental data points, is recommended:

$Nu=0.023\operatorname{Re}_{\ell 0}^{0.8}\Pr _{\ell }^{0.4}\left[ {{(1-x)}^{0.8}}+\frac{3.8{{x}^{0.76}}{{(1-x)}^{0.04}}}{{{({{p}_{sat}}/{{p}_{c}})}^{0.38}}} \right]\qquad\qquad(1)$

where ${{\operatorname{Re}}_{\ell o}}=\frac{\dot{{m}''}D}{{{\mu }_{\ell }}}$ (see Correlations Based on the Separated Flow Model) and ${{\Pr }_{\ell }}$ is the liquid Prandtl number. Equation (1) was obtained by modifying the Dittus-Boelter equation for single-phase forced convection in a tube. It is applicable to the following conditions: $0.002\le {{p}_{sat}}/{{p}_{c}}\le 0.44,$ $21{{\text{ }}^{\text{o}}}\text{C}\le {{T}_{sat}}\le 310{{\text{ }}^{\text{o}}}\text{C},$ $3\text{m/s}\le {{w}_{v}}\le 300\text{m/s},$ $0\le x\le 1$ , $10.8\text{ kg/}{{\text{m}}^{\text{2}}}\text{-s}\le \dot{{m}''}\le 1599\text{ kg/}{{\text{m}}^{\text{2}}}\text{-s},$ ${{\operatorname{Re}}_{\ell 0}}\ge 350,$ and ${{\Pr }_{\ell }}>0.5.$ Moser et al. (1998) verified eq. (1) using 1197 data points for 6 different kinds of refrigerants and concluded that the deviation of eq. (1) is 14.37%.

Figure 1: Cross-section of stratified flow.

At lower vapor velocity, interfacial shear force becomes less important and gravity becomes the dominant force in convective condensation in the tube. As shown in Fig. 1, due to gravitational force, the condensate in the top portion of the tube flows into the bottom portion, and the two-phase flow lies in the stratified regime. The stratified regime is very common for low mass flow rates or short tubes.

The average heat transfer coefficient for condensation in a tube with stratified flow is

$\bar{h}=\frac{{{\theta }_{strat}}}{\pi }{{h}_{d}}+\left( 1-\frac{{{\theta }_{strat}}}{\pi } \right){{h}_{b}}\qquad\qquad(2)$

where $h d$ and $h b$ are heat transfer coefficients of the drainage condensate and the condensate at the bottom. Since condensation occurs primarily at the top portion of the tube where the liquid layer is relatively thin, the condensation occurring on the surface of the stratified liquid can be neglected ( ${{h}_{b}}\ll {{h}_{d}}$ ). Thus, eq. (2) is simplified as

$\bar{h}=\frac{{{\theta }_{strat}}}{\pi }{{h}_{d}}\qquad\qquad(3)$

Since condensation occurring in the drainage condensate is film condensation, one can expect that the heat transfer coefficient can be obtained using a correlation similar to the Nusselt solution.

${{h}_{d}}=F({{\theta }_{strat}}){{\left[ \frac{g{{\rho }_{\ell }}({{\rho }_{\ell }}-{{\rho }_{v}})k_{\ell }^{3}{{{{h}'}}_{\ell v}}}{{{\mu }_{\ell }}D({{T}_{sat}}-{{T}_{w}})} \right]}^{1/4}}\qquad\qquad(4)$

where the function $F (θ s t r a t)$ can be obtained by (Jaster and Kosky, 1976)

$F({{\theta }_{strat}})=0.728\frac{\pi }{{{\theta }_{strat}}}{{\left[ \frac{\pi }{{{\theta }_{strat}}}+\frac{\sin 2(\pi -{{\theta }_{strat}})}{2\pi } \right]}^{3/4}}\qquad\qquad(5)$

The void fraction, $α$ , is related to the stratification angle, $θ s t r a t$ , by

$\alpha =\frac{\pi }{{{\theta }_{strat}}}+\frac{\sin 2(\pi -{{\theta }_{strat}})}{2\pi }\qquad\qquad(6)$

Substituting eqs. (5) and (6) into eq. (4) yields

${{h}_{d}}=0.728\frac{\pi }{{{\theta }_{strat}}}{{\alpha }^{3/4}}{{\left[ \frac{g{{\rho }_{\ell }}({{\rho }_{\ell }}-{{\rho }_{v}})k_{\ell }^{3}{{{{h}'}}_{\ell v}}}{{{\mu }_{\ell }}D} \right]}^{1/4}}\qquad\qquad(7)$

The average heat transfer coefficient can be obtained by substituting eq. (7) into eq. (3), i.e.,

$\bar{h}=0.728{{\alpha }^{3/4}}{{\left[ \frac{g{{\rho }_{\ell }}({{\rho }_{\ell }}-{{\rho }_{v}})k_{\ell }^{3}{{{{h}'}}_{\ell v}}}{{{\mu }_{\ell }}D} \right]}^{1/4}}\qquad\qquad(8)$

The condition for which eq. (8) applies is ${{\operatorname{Re}}_{vo}}\le 3.5\times {{10}^{4}}$ . The void fraction in eq. (8) can be obtained by

$\alpha ={{\left[ 1+\frac{1-x}{x}{{\left( \frac{{{\rho }_{v}}}{{{\rho }_{\ell }}} \right)}^{2/3}} \right]}^{-1}}\qquad\qquad(9)$

In addition to eqs. (1) and (8), Dobson and Chato (1998) proposed the empirical correlations that took into account the effect of different flow regimes. For annular flow and $\dot{{m}''}\ge 500\text{ kg/}{{\text{m}}^{\text{2}}}\text{-s}$ , Dobson and Chato (1998) recommended the following correlation:

$\text{Nu}=\frac{hD}{{{k}_{\ell }}}=0.023\operatorname{Re}_{\ell }^{0.8}\Pr _{\ell }^{0.4}\left( 1+\frac{2.22}{X_{tt}^{0.89}} \right)\qquad\qquad(10)$

where $X t t$ is the Martinelli parameter for turbulent flow in both liquid and vapor phases, i.e.,

${{X}_{tt}}={{\left( \frac{1-x}{x} \right)}^{0.9}}{{\left( \frac{{{\rho }_{v}}}{{{\rho }_{\ell }}} \right)}^{0.5}}{{\left( \frac{{{\mu }_{\ell }}}{{{\mu }_{v}}} \right)}^{0.1}}\qquad\qquad(11)$

When the flow pattern is stratified wavy flow, the empirical correlation is

$\text{Nu}=\frac{hD}{{{k}_{\ell }}}=\frac{0.23\operatorname{Re}_{vo}^{0.12}}{1+1.11X_{tt}^{0.58}}{{\left[ \frac{G{{a}_{\ell }}{{\Pr }_{\ell }}}{J{{a}_{\ell }}} \right]}^{0.25}}+\left( 1-\frac{{{\theta }_{strat}}}{\pi } \right)N{{u}_{strat}}\qquad\qquad(12)$

where the first term on the right-hand side of eq. (12) accounts for the contribution of the film condensation on the top portion of the tube. The contribution of the stratified liquid is accounted for by the second term on the right-hand side of eq. (12). $\text{G}{{\text{a}}_{\ell }}$ is the Galileo number for the liquids defined as

$\text{G}{{\text{a}}_{\ell }}=\frac{g{{D}^{3}}}{\nu _{\ell }^{2}}\qquad\qquad(13)$

The angle from the top of the tube to the stratified liquid layer in the bottom $θ s t r a t$ is estimated by

${{\theta }_{strat}}=\pi -\arccos (2\alpha -1)\qquad\qquad(14)$

where the void fraction is approximated by $\alpha =\frac{1}{1+\frac{1-x}{x}\frac{{{\rho }_{v}}}{{{\rho }_{\ell }}}S}$ (see Void Fraction Model for Two-Phase Flow), with the slip ratio $S={{\left( {{\rho }_{\ell }}/{{\rho }_{v}} \right)}^{1/3}},$ i.e.,

$\alpha =\frac{x}{x+(1-x){{\left( {{\rho }_{v}}/{{\rho }_{\ell }} \right)}^{2/3}}}\qquad\qquad(15)$

which is referred to as the Zivi void fraction equation (Zivi, 1964).

The Nusselt number for forced convective heat transfer of the stratified liquid, $Nu s t r a t$ , in eq. (12) is

$\text{N}{{\text{u}}_{strat}}=0.0195\operatorname{Re}_{\ell }^{0.8}\Pr _{\ell }^{0.4}{{\left( 1.376+\frac{{{c}_{1}}}{X_{tt}^{{{c}_{2}}}} \right)}^{1/2}}\qquad\qquad(16)$

where

${{c}_{1}}=\left\{ \begin{matrix} 4.172+5.48F{{r}_{\ell }}-1.564Fr_{\ell }^{2} & F{{r}_{\ell }}\le 0.7 \\ 7.242 & F{{r}_{\ell }}>0.7 \\ \end{matrix} \right.\qquad\qquad(17)$

and

${{c}_{2}}=\left\{ \begin{matrix} 1.773-0.169F{{r}_{\ell }} & F{{r}_{\ell }}\le 0.7 \\ 1.655 & F{{r}_{\ell }}>0.7 \\ \end{matrix} \right.\qquad\qquad(18)$

The transition from annular flow to stratified flow is characterized by the modified Froude transition number, which is defined as

$\text{F}{{\text{r}}_{so}}={{c}_{3}}\operatorname{Re}_{\ell }^{{{c}_{4}}}{{\left( \frac{1+1.09X_{tt}^{0.039}}{{{X}_{tt}}} \right)}^{1.5}}\qquad\qquad(19)$

where $c 3 = 0.025 and c 4 = 1.59$ for ${{\operatorname{Re}}_{\ell }}\le 1250$ and $c 3 = 1.26 and c 4 = 1.04$ for ${{\operatorname{Re}}_{\ell }}>1250$ . Dobson and Chato (1998) suggested that eq. (12) is applicable when $\dot{{m}''}<500\text{ kg/}{{\text{m}}^{\text{2}}}\text{-s}$ and $Fr s o < 20$ .

References

Dobson, M.K., and Chato, J.C., 1998, “Condensation in Smooth Horizontal Tubes,” ASME Journal of Heat Transfer, Vol. 120, pp. 193-213.

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA

Jaster, H., and Kosky, P.G., 1976, “Condensation Heat Transfer in a Mixed Flow Regime,” International Journal of Heat Transfer, Vol. 19, pp. 95-99.

Moser, K.W., Webb, R.L., and Na, B., 1998, “A New Equivalent Reynolds Number Model for Condensation in Smooth Tubes,” ASME Journal of Heat Transfer, Vol. 120, pp. 410-417.

Shah, M.M., 1979, “A General Chart Correlation for Heat Transfer during Condensation Inside Pipes,” International Journal of Heat and Mass Transfer, Vol. 22, pp. 547-556.

Zivi, S.M., 1964, “Estimation of Steady-State Void Fraction by Means of the Principle of Minimum Energy Production,” ASME Journal of Heat Transfer, Vol. 86, pp. 247-252.

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Heat transfer predictions for forced convective condensation

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