Discussion: “Heat Transfer and Wall Heat Flux Partitioning During Subcooled Flow Nucleate Boiling–A Review” (Warrier, G.R., and Dhir, V.K., 2006, Journal of Heat Transfer, 128, pp. 1243–1256)

Kandlikar, Satish G.

doi:10.1115/1.2745830

In their paper (1), the authors state on p. 1245 that Boyd and Meng (2) in 1996 suggested an interpolation method for calculating the heat transfer characteristics in the partial boiling region, along with Eqs. (13)-(14). The authors of (1) further state that Kandlikar (3) in 1998 proposed a similar scheme and that the constants $a$ ⁠, $b$ ⁠, and $m$ were assumed to be constant.

In this discussion, we would like to point out a few errors/omissions in the above referenced paper (1).

1. In their technical note, Boyd and Meng (2) refer to the paper by Kandlikar (4), published in 1990, for the Eqs. (13)-(14) referred to in (1). Unfortunately, Boyd and Meng quoted the wrong reference for the Kandlikar work in which the original model and equations were reported. The 1990 paper by Kandlikar (4), erroneously referred by Boyd and Meng, presents a correlation in the saturated flow boiling region and makes no reference to subcooled flow boiling. The correct reference in Boyd and Meng’s paper should have been Kandlikar (5), which was published in 1991.

2. In his 1991 paper, Kandlikar (5) presented Eqs. (21)-(29) with an accompanying Fig. 4 explaining the construction in the subcooled partial boiling region. Immediately following Eq. (26), a note appears regarding the exponent $m$ ⁠, which matches the slope at the two ends. Kandlikar (5), however, contains typographical errors in Eqs. (25) and (26) for coefficients $a$ and $b$ ⁠.

3. Those typographical errors were later corrected by Kandlikar (3) in 1998.

4. Another error that appears in (1) is the incorrect year of publication of the paper by Boyd and Meng (2), cited as Ref. [12]; the correct year of publication is 1995, not 1996.

5. Discussion related to Fig. 6 appearing in (1) is correct, except that the 1998 paper by Kandlikar (3) referred therein simply corrects the typographical error and provides a more detailed comparison with the available experimental data.

6. Equation (21) appearing in Warrier and Dhir (1) is based on the paper by Kandlikar (3) as correctly reported by the authors of (1). However, the equation itself is incorrectly reproduced. The correct form of the equation (appearing as Eq. 14 in Ref (3)) is as follows:

\dot{q} = {[1058.0 {(\dot{m} h_{f g})}^{- 0.7} F_{Fl} h_{LO} Δ T_{sat}]}^{1 ∕ 0.3}

1

where

\dot{m}

is the mass flux

(kg ∕ m^{2} s)

⁠. The negative sign in the exponent 0.7 is missing in Eq. (21) in (1), and the wall superheat

Δ T_{sat}

is incorrectly replaced by

(Δ T_{w} + Δ T_{sub})

⁠.

Subcooled Flow Boiling Model Description

For convenience, the correct form of the subcooled flow boiling model and coefficients are presented below.

Figure 1 shows a schematic representation of the subcooled flow boiling curve extending from the single-phase region at point C to the fully developed boiling at point E. In the single-phase region, to the left of C in Fig. 1, heat flux

\dot{q}

is given by the following equation:

\dot{q} = h_{LO} (Δ T_{sat} + Δ T_{sub})

2

where

h_{LO}

is the single-phase heat transfer coefficient with all flow as liquid, the local wall superheat is

Δ T_{sat} (= T_{w} - T_{sat})

and the local liquid subcooling is

Δ T_{sub} (= T_{sat} - T_{f})

⁠, the local saturation temperature is

T_{sat}

⁠, the wall temperature is

T_{w}

and the liquid temperature is

T_{f}

⁠.

Figure 1

A representative log-log plot of q̇ versus ΔTsat for describing the Kandlikar model in the partial boiling region, adapted from Kandlikar (5)

View large Download slide

A representative log-log plot of $\dot{q}$ versus $Δ T_{sat}$ for describing the Kandlikar model in the partial boiling region, adapted from Kandlikar (5)

In the fully developed boiling region, the heat transfer rate is related to the local wall superheat by the following equation:

\frac{\dot{q}}{Δ T_{sat}} = 1058.0 {Bo}^{0.7} F_{Fl} h_{LO}

3

where Bo is the boiling number

(\dot{q} ∕ (G h_{f g}))

G

is the total mass flux

(kg ∕ m^{2} s)

⁠, and

h_{f g}

is the latent heat of vaporization (J/kg), and

F_{Fl}

is the fluid-dependent parameter in the Kandlikar correlation (3). The value of

F_{Fl}

is 1 for water and all other fluids flowing in stainless steel tubes. For specific fluids in different tube materials, refer to (4) or other more recent publications.

The equation for the

\dot{q} - Δ T_{sat}

plot in the partial boiling region, the main focus of the current discussion, is given by the following equation:

\dot{q} = a + b {(T_{w} - T_{sat})}^{m} = a + b {(Δ T_{sat})}^{m}

4

The constants

a

⁠,

b

⁠, and

m

are functions of heat flux

\dot{q}

⁠. The slope of the heat flux versus wall superheat in the partial boiling region is matched with the two limiting values, i.e.,

m = 1

in the single-phase region at the beginning of the partial boiling region, identified by point C, and

m = 1 ∕ 0.3

at the beginning of the fully developed boiling region identified by point E. Thus, the values of

a

and

b

are obtained in terms of the heat fluxes and wall superheats at C and E, and the value of

m

is obtained in terms of the heat fluxes at C, E and at the desired location, where heat flux is

\dot{q}

and wall superheat is

Δ T_{sat}

⁠.

b = \frac{{\dot{q}}_{E} - {\dot{q}}_{C}}{{(Δ T_{sat, E})}^{m} - {(Δ T_{sat, C})}^{m}}

5

and

a = {\dot{q}}_{C} - b {(Δ T_{sat, C})}^{m}

6

Note that there were typographical errors in Kandlikar (5) that erroneously omitted the exponent

m

in Eqs. ,.

The value of

m

depends on the heat flux, and is allowed to vary linearly from

m = 1

at C to

m = 1 ∕ 0.3

at D. Thus,

m = n + p \dot{q}

7

and the values of

n

and

p

are obtained as follows:

p = (1 ∕ 0.3 - 1) ∕ ({\dot{q}}_{E} - {\dot{q}}_{C})

8

and

n = 1 - p {\dot{q}}_{C}

9

Note that

n

and

p

are constants for a system (for a given geometry and operating conditions), whereas the values of

m

⁠,

a

⁠, and

b

depend of the local value of

\dot{q}

⁠. Thus, the value of

Δ T_{sat}

can be obtained directly from a known value of

\dot{q}

⁠, while an iterative scheme is needed to calculate

\dot{q}

for a given value of

Δ T_{sat}

in this region.

Further details on calculating ${\dot{q}}_{C}$ and ${\dot{q}}_{E}$ are given in Kandlikar (3).

1.

Warrier

,

G. R.

, and

Dhir

,

V. K.

, 2006, “

Heat Transfer and Wall Heat Flux Partitioning During Subcooled Flow Nucleate Boiling–A Review

,”

ASME J. Heat Transfer

0022-1481,

128

, pp.

1243

–

1256

.

Crossref

2.

Boyd

,

R. D.

, and

Meng

,

X.

, 1995, “

Boiling Curve Correlation for Subcooled Flow Boiling

,”

Int. J. Heat Mass Transfer

0017-9310,

38

, pp.

758

–

760

.

Crossref

3.

Kandlikar

,

S. G.

, 1998, “

Heat Transfer Characteristics in Partial Boiling, Fully Developed Boiling, and Significant Void Flow Regions of Subcooled Flow Boiling

,”

ASME J. Heat Transfer

0022-1481,

120

, pp.

390

–

401

.

4.

Kandlikar

,

S. G.

, 1990, “

A General Correlation for Two-phase Flow Boiling Heat Transfer Inside Horizontal and Vertical Tubes

,”

ASME J. Heat Transfer

0022-1481,

112

, pp.

219

–

228

.

Crossref

5.

Kandlikar

,

S. G.

, 1991, “

Development of a Flow Boiling Map for Subcooled and Saturated Flow Boiling of Different Fluids Inside Circular Tubes

,”

ASME J. Heat Transfer

0022-1481,

113

, pp.

190

–

200

.

Crossref

by American Society of Mechanical Engineers

Discussion: “Heat Transfer and Wall Heat Flux Partitioning During Subcooled Flow Nucleate Boiling–A Review” (Warrier, G.R., and Dhir, V.K., 2006, Journal of Heat Transfer, 128, pp. 1243–1256)

Subcooled Flow Boiling Model Description

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Discussion: “Heat Transfer and Wall Heat Flux Partitioning During Subcooled Flow Nucleate Boiling–A Review” (Warrier, G.R., and Dhir, V.K., 2006, Journal of Heat Transfer, 128, pp. 1243–1256)

Subcooled Flow Boiling Model Description

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