# Heat Transfer Predictions for Forced Convective Boiling

The different flow regimes have significant effects on the heat transfer characteristics of convective boiling in a tube. While heat transfer for subcooled liquid and superheated vapor can be easily handled by correlations for single-phase heat transfer, the intermediate heat transfer mechanism is complicated by phase change from liquid to vapor. After boiling is initiated in the tube, vapor bubbles are generated at certain nucleate sites while the rest of the inner surface of the tube remains in contact with the liquid. Under these conditions, the heat transfer mechanism is a combination of two parallel processes: single-phase convection in the liquid and nucleate boiling.

The overall heat transfer coefficient for convective boiling in an upward vertical tube can be written as (Chen, 1963)

$h=F{{h}_{\ell }}+S{{h}_{b}} \qquad \qquad(1)$

where ${{h}_{\ell }}$ and hb are heat transfer coefficients for single-phase convection of the liquid and nucleate boiling, respectively. F and S in eq. (1) are dynamic factors that modify the contributions of single-phase liquid convection and nucleate boiling, respectively

The single-phase heat transfer coefficient for liquid alone can be obtained by using the Dittus-Boelter/McAdams equation:

${{h}_{\ell }}=0.023\left( \frac{{{k}_{\ell }}}{D} \right)\operatorname{Re}_{\ell }^{0.8}\Pr _{\ell }^{0.4} \qquad \qquad(2)$

The contribution of nucleate boiling is determined by using the correlation proposed by Forster and Zuber (1955) for pool boiling:

${{h}_{b}}=0.00122\left[ \frac{k_{\ell }^{0.79}c_{p,\ell }^{0.45}\rho _{\ell }^{0.49}}{{{\sigma }^{0.5}}\mu _{\ell }^{0.29}h_{fg}^{0.24}\rho _{v}^{0.24}} \right]{{\left[ {{T}_{w}}-{{T}_{sat}}({{p}_{\ell }}) \right]}^{0.24}}{{\left[ {{p}_{sat}}({{T}_{w}})-{{p}_{\ell }} \right]}^{0.75}} \qquad \qquad(3)$

The convective boiling factor F can be obtained by a regression of experimental data (Chen, 1963):

$F=\left\{ \begin{matrix} 1 & X_{tt}^{-1}\le 0.1 \\ 2.35{{(0.213+X_{tt}^{-1})}^{0.736}} & X_{tt}^{-1}>0.1 \\ \end{matrix} \right. \qquad \qquad(4)$

where Xtt is the Lockhart-Martinelli parameter obtained by (see Heat Transfer Predictions for Forced Convective Condensation)

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

The nucleate boiling suppression factor S is

$S=\frac{1}{1+2.53\times {{10}^{-6}}\operatorname{Re}_{TP}^{1.17}} \qquad \qquad(5)$

where

${{\operatorname{Re}}_{TP}}={{\operatorname{Re}}_{\ell }}{{F}^{1.25}} \qquad \qquad(6)$

is the local two-phase Reynolds number.

Based on more than 10,000 experimental data points for different fluids, including water, refrigerants, and cryogents, Kandlikar (1990, 1991) proposed the following generalized heat transfer correlation for convective boiling in both vertical and horizontal tubes:

$h=\max \left[ \begin{matrix} {{h}_{NBD,}} & {{h}_{CBD}} \\ \end{matrix} \right] \qquad \qquad(7)$

where hNBD and hCBD are the nucleate boiling dominant and convective boiling dominant heat transfer coefficients, and they are obtained by

${{h}_{NBD}}=\left[ 0.6683\text{C}{{\text{o}}^{-0.2}}{{f}_{2}}(\text{F}{{\text{r}}_{\ell o}})+1058\text{B}{{\text{o}}^{0.7}}{{F}_{f\ell }} \right]{{(1-x)}^{0.8}}{{h}_{\ell o}} \qquad \qquad(8)$
${{h}_{CBD}}=\left[ 1.136\text{C}{{\text{o}}^{-0.9}}{{f}_{2}}(\text{F}{{\text{r}}_{\ell o}})+667.2\text{B}{{\text{o}}^{0.7}}{{F}_{f\ell }} \right]{{(1-x)}^{0.8}}{{h}_{\ell o}} \qquad \qquad(9)$

where Co is the convective number, Bo is the boiling number, and $\text{F}{{\text{r}}_{\ell o}}$ is the Froude number.

$\text{Co}={{\left( \frac{{{\rho }_{v}}}{{{\rho }_{\ell }}} \right)}^{0.5}}{{\left( \frac{1-x}{x} \right)}^{0.8}} \qquad \qquad(10)$
$\text{Bo}=\frac{{{{{q}''}}_{w}}}{A\dot{{m}''}{{h}_{\ell v}}} \qquad \qquad(11)$
$\text{F}{{\text{r}}_{\ell o}}=\frac{{{{\dot{{m}''}}}^{2}}}{\rho _{\ell }^{2}gD} \qquad \qquad(12)$

The Froude number multiplier, ${{f}_{2}}(\text{F}{{\text{r}}_{\ell o}})$, is

${f_2}({\rm{F}}{{\rm{r}}_{\ell o}}) = \left\{ {\begin{array}{*{20}{c}} 1 & {{\rm{vertical or horizontal tubes (F}}{{\rm{r}}_{\ell o}} \ge {\rm{0}}{\rm{.04)}}} \\ {{{(25{\rm{F}}{{\rm{r}}_{\ell o}})}^{0.3}}} & {{\rm{horizontal tubes (F}}{{\rm{r}}_{\ell o}}{\rm{ < 0}}{\rm{.04) }}} \\ \end{array}} \right.\qquad\qquad(13)$

For liquids with a Prandtl number between 0.5 and 2000, a range that covers most fluids except liquid metal, the single-phase all-liquid heat transfer coefficient ${{h}_{\ell o}}$ in eqs. (8) and (9) is

${{h}_{\ell o}}=\left\{ \begin{matrix} \frac{{{k}_{\ell }}}{D}\frac{({{\operatorname{Re}}_{\ell o}}-1000){{\Pr }_{\ell }}(f/2)}{1+12.7(\Pr _{\ell }^{2/3}-1){{(f/2)}^{0.5}}} & 2300\le {{\operatorname{Re}}_{\ell o}}\le {{10}^{4}} \\ \frac{{{k}_{\ell }}}{D}\frac{{{\operatorname{Re}}_{\ell o}}{{\Pr }_{\ell }}(f/2)}{1.07+12.7(\Pr _{\ell }^{2/3}-1){{(f/2)}^{0.5}}} & {{10}^{4}}\le {{\operatorname{Re}}_{\ell o}}\le 5\times {{10}^{6}} \\ \end{matrix} \right. \qquad \qquad(14)$

where the friction factor f in eq. (14) is given by

$f={{\left( 1.58\ln {{\operatorname{Re}}_{\ell o}}-3.28 \right)}^{-2}} \qquad \qquad(15)$
Table 1: ${{F}_{f\ell }}$ for copper tube
 Fluid ${F_{f \ell}}$ Fluid ${F_{f \ell}}$ Water 1.00 R-114 1.24 R-11 1.30 R-134a 1.63 R-12 1.50 R-152a 1.10 R-13B1 1.31 Nitrogen 4.70 R-22 2.20 Neon 3.50 R-113 1.30

The fluid-surface parameter ${{F}_{f\ell }}$ depends on the combination of the liquid and tube material used. For stainless steel tubing, ${{F}_{f\ell }}$ is taken as 1 regardless of the type of fluid. For copper tubing, ${{F}_{f\ell }}$ can be obtained from Table 1 Kandlikar (1990; 1991). The above correlation is valid for $0.001\le x\le 0.95.$

Under constant heat flux conditions, the wall temperature will sharply increase when the heat transfer mechanism inside the tube suddenly changes from two-phase to single vapor phase heat transfer. Since the heat transfer inside the tube always begins with the single liquid phase, then changes to convective boiling, and finally to single phase vapor, the critical heat flux (CHF) phenomenon always occurs at a point near the exit of the tube.

For this reason, CHF for convective boiling inside the tube is a local phenomenon. There are numerous empirical correlations available in the literature. The most generalized model to predict CHF for forced convective boiling is recommended by Katto and Ohno (1984).

${{q''}_{max}} = {{q''}_{co}}(1 + {K_K} \frac { {h_{\ell, sat}} - {h_{in}}} {{h_{\ell v}}} )\qquad\qquad(16)$

For $\gamma ={{\rho }_{v}}/{{\rho }_{\ell }}<0.15$

${{q''}_{co}} = \frac{ {{q''}_{co1}} \qquad {{q''}_{co1}}< {{q''}_{co2}} } {min({{q''}_{co2}},{{q''}_{co3}})\qquad {{q''}_{co1}}>{{q''}_{co2}} } \qquad \qquad (17)$
${K_K} = max ({K_{K1}}, {K_{K2}}) \qquad\qquad(18)$

For $\gamma ={{\rho }_{v}}/{{\rho }_{\ell }}>0.15$

${{q''}_{co}} = ( \frac{ {{q''}_{co1}} \qquad {{q''}_{co1}}< {{q''}_{co5}} } {min({{q''}_{co4}},{{q''}_{co5}})\qquad {{q''}_{co1}}>{{q''}_{co5}} } \qquad \qquad (19)$
${K_K} = ( \frac {{K_{K1}} \qquad {K_{K1}} > {K_{K2}} } {min({K_{K2}}, {K_{K3}}) {K_{K1}} <= {K_{K2}} } \qquad\qquad(20)$

where

${{{q}''}_{\text{co1}}}={{C}_{K}}\dot{{m}''}{{h}_{\ell v}}\text{We}_{k}^{-0.043}{{\left( \frac{L}{D} \right)}^{-1}} \qquad \qquad(20)$

${{{q}''}_{\text{co2}}}=0.10\dot{{m}''}{{h}_{\ell v}}{{\gamma }^{\text{1}\text{.33}}}\text{We}_{k}^{-1/3}\left[ \frac{1}{1+0.0031(L/D)} \right] \qquad \qquad(21)$

${{{q}''}_{\text{co3}}}=0.098\dot{{m}''}{{h}_{\ell v}}{{\gamma }^{\text{1}\text{.33}}}\text{We}_{k}^{-0.433}\left[ \frac{{{(L/D)}^{0.27}}}{1+0.0031(L/D)} \right] \qquad \qquad(22)$

${{{q}''}_{\text{co}4}}=0.0384\dot{{m}''}{{h}_{\ell v}}{{\gamma }^{\text{0}\text{.6}}}\text{We}_{k}^{-0.173}\left[ \frac{1}{1+0.280\text{We}_{k}^{-0.233}(L/D)} \right] \qquad \qquad(23)$

${{{q}''}_{co5}}=0.234\dot{{m}''}{{h}_{\ell v}}{{\gamma }^{\text{0}\text{.513}}}\text{We}_{k}^{-0.433}\left[ \frac{{{(L/D)}^{0.27}}}{1+0.0031(L/D)} \right] \qquad \qquad(24)$

${{K}_{K1}}=\frac{1.043}{4{{C}_{K}}\text{We}_{k}^{-0.043}} \qquad \qquad(25)$

${{K}_{K2}}=\left( \frac{5}{6} \right)\frac{0.0124+D/L}{{{\gamma }^{\text{1}\text{.33}}}\text{We}_{k}^{-1/3}} \qquad \qquad(26)$

${{K}_{K3}}=1.12\frac{1.52\text{We}_{k}^{-0.233}+D/L}{{{\gamma }^{\text{0}\text{.6}}}\text{We}_{k}^{-0.173}} \qquad \qquad(27)$

$\gamma =\frac{{{\rho }_{v}}}{{{\rho }_{\ell }}} \qquad \qquad(28)$

$\text{W}{{\text{e}}_{k}}=\frac{{{{\dot{{m}''}}}^{2}}L}{{{\rho }_{\ell }}\sigma } \qquad \qquad(29)$

${{C}_{K}}=\left\{ \begin{matrix} 0.25\text{ L/D}<\text{50} \\ 0.25+0.009\left[ L/D-50 \right]\text{ 50}\le \text{L/D}\le \text{150} \\ 0.34\text{ L/D}>\text{150} \\ \end{matrix}\text{ } \right.\text{ } \qquad \qquad(30)$

The critical heat flux predicted using eq. (16) agreed reasonably well with a variety of fluids, including water, ammonia, benzene, ethanol, helium, hydrogen, nitrogen, R-12, R-21, R-22, R-113, and potassium.

Heat transfer coefficients for boiling in both vertical and horizontal tubes are often measured from experiments using electrical heating that result in axially and circumferentially uniform heat flux. While this approach can give reasonable boundary conditions for boiling in a vertical tube, Thome (2004) pointed out that electrical heating for boiling in a horizontal tube is not preferred, because circumferential conduction in the tube wall from the hot, dry-wall condition at the top to the colder, wet-wall condition at the bottom yields unknown boundary conditions. Therefore, countercurrent hot water heating that can provide reasonable boundary conditions is preferred.

## References

Chen, I.C., 1963, “A Correlation for Boiling Heat Transfer to Saturated Fluids in Convective Flow,” ASME preprint 63-HT-34, Presented at 6th National Heat Transfer Conference, Boston, MA.

Forster, H.K., and Zuber, N., 1955, “Dynamics of Vapor Bubbles and Boiling Heat Transfer,” AIChE Journal, Vol. 1, pp. 531-535.

Kandlikar, S.G., 1990, “A General Correlation for Two-Phase Flow Boiling Heat Transfer Coefficient Inside Horizontal and Vertical Tubes,” ASME Journal of Heat Transfer, Vol. 112, pp. 219-228.

Kandlikar, S.G., 1991, “Development of a Flow Boiling Map for Subcooled and Saturated Flow Boiling of Different Fluids in Circular Tubes,” ASME Journal of Heat Transfer, Vol. 113, pp. 190-200.

Katto, Y., and Ohno, H., 1984, “An Improved Version of the Generalized Correlation of Critical Heat Flux for the Forced Convection Boiling in Uniformly Heated Vertical Tubes,” International Journal of Heat and Mass Transfer, Vol. 27, pp. 1641-1648.

Thome, J.R., 2004, Engineering Data Book III, Wolverine Tube, Inc., Huntsville, AL.