A simplified model of direct-contact heat transfer in desalination system utilizing LNG cold energy

Abstract

With the increasingly extensive utilization of liquefied natural gas (LNG) in China today, sustainable and effective using of LNG cold energy is becoming increasingly important. In this paper, the utilization of LNG cold energy in seawater desalination system is proposed and analyzed. In this system, the cold energy of the LNG is first transferred to a kind of refrigerant, i.e., butane, which is immiscible with water. The cold refrigerant is then directly injected into the seawater. As a result, the refrigerant droplet is continuously heated and vaporized, and in consequence some of the seawater is simultaneously frozen. The formed ice crystal contains much less salt than that in the original seawater. A simplified model of the direct-contact heat transfer in this desalination system is proposed and theoretical analyses are conducted, taking into account both energy balance and population balance. The number density distribution of two-phase bubbles, the heat transfer between the two immiscible fluids, and the temperature variation are then deduced. The influences of initial size of dispersed phase droplets, the initial temperature of continuous phase, and the volumetric heat transfer coefficient are also clarified. The calculated results are in reasonable agreement with the available experimental data of the R114/water system.

Keywords

liquefied natural gas (LNG) / cold energy utilization / desalination / direct-contact heat transfer

Introduction

As a type of energy resource that is relatively clean and convenient for transportation and storage, the global liquefied natural gas (LNG) demand, including the liquefaction of the offshore associated gas [1-3] and coalmine gas (coal-bed methane) [4,5], has grown rapidly and the world wide LNG trade has increased steadily in recent years, particularly in the Asia Pacific region [6-9]. LNG will release a huge amount of cold energy, approximately 830 kJ/kg, during its re-gasification processes [10]. As a result, it is beneficial to energy saving and environmental conservation to recover and use such cold energy effectively. In addition, many LNG receiving terminals are located in the seashore where there is usually a serious lack of fresh water. Forasmuch, it is reasonable to utilize the cold energy of LNG for desalination, thus supplying fresh water for the LNG receiving terminal and even other places where fresh water are need.

A few of tentative studies of the desalination system utilizing LNG cold energy were conducted. Cravalho et al. [11] proposed a net zero work system, utilizing the cold energy of LNG during vaporization. Antonelli [12] presented a process named ORC+SRF, which conceived that organic Rankine cycle (ORC) process drove the secondary refrigerant freezing (SRF) process. However, because of the idealization and complexity of the above systems, it is difficult to be applied practically in industry. Accordingly, it seems that the SRF process is much more feasible in practice. In the SRF process, the secondary refrigerant, e.g., butane (R600), are utilized to absorb cold energy from the LNG in cryogenic heat exchanger, and then to absorb heat from the seawater through its evaporation process, aimed to obtain ice crystals.

The mechanism of direct contact heat transfer which is the basis of the process of freezing desalination has been studied by many researchers. Sideman et al. [13-16] conducted a systematic study on the transfer characteristics of a single immiscible refrigerant droplet while rising in continuous phase, such as water and seawater. Tochitani et al. [17,18] investigated the vaporization process of a single pentane of furan droplet in high viscosity continuous phase. Raina et al. [19,20] introduced the effect of viscous shear on the spreading of dispersed liquid over bubble surface, then considered the effect of sloshing and finally obtained the correlation of the heat transfer coefficient. Raina et al. [21] also dealt with the motion of vaporizing two-phase bubble, and got the correlation of the instantaneous velocity of the rising two-phase bubble, which was confirmed by many experimental data. Battya et al. [22] presented the correlation of Nusselt number of single droplet, which was applicable to any immiscible dispersed and continuous phase.

For the purpose of practical application, some researchers investigated the multiple droplets undergoing vaporization in immiscible continuous phase liquid. Sideman et al. [23,24] investigated the transfer characteristics of spray column, in which volatile dispersed phase evaporated while rising in immiscible continuous phase. Seetharamu et al. [25] studied the effects of operational parameters, such as operating column height, the temperature difference, the dispersed phase flow rate, and the diameter and number of orifices in the distributor. Mori [26] presented an analytic model of direct contact heat transfer of the spray column, and obtained an expression of the volumetric heat transfer coefficient which was an important parameter in the system. On the background of ice thermal energy storage, Kiatsiriroat et al. [27,28] used a lumped model to analyze the characteristics of the phenomenon. Byrd et al. [29], Core and Mulligan [30] and Song et al. [31-33] introduced population balance formulation of two phase bubbles, and solved the problem with different mathematic methods. Based on the above investigations, it seemed much more reasonable to consider the population balance of bubbles, rather than to assume the homogeneous of the system.

In the present study, a simplified mathematical-physical model of direct contact heat transfer in desalination system utilizing LNG cold energy is established and analyzed. The characteristics of direct contact heat transfer are systematically investigated taking both the energy balance and population balance into consideration. The influences of parameters on the initial dispersed phase droplets are clarified, and the volumetric heat transfer coefficient, which is the main design parameter of this direct contact heat transfer system, is also obtained.

Mathematical-physical model

The direct contact heat transfer process for the proposed desalination system is simplified into a physical model, as illustrated in Fig. 1. The dispersed phase liquid is injected into the spray column, which is filled with stagnant continuous phase liquid, through a distributor at the bottom at a saturated temperature of the dispersed phase. As the temperature of the continuous phase is higher than that of the dispersed phase and the density of the dispersed phase lower than that of the continuous phase, the dispersed phase droplet evaporates and arises in the column, and finally exhausts on the top of the column. On the other hand, the temperature of the continuous phase liquid decreases.

Population balance equation

As the population balance equation is a powerful mathematical tool to analyze particle dispersions, it is adopted in this work to investigate the distribution of the dispersed two-phase bubbles. There are several approaches to solve the population balance equation, and the method of classes, proposed by Marchal et al. [34] seems to be convenient to deal with this problem.

Let {R₀, R₁, …, R_M} be a set of dispersed phase bubble radius, whereR₀ is the smallest size of these bubbles. In this case, R₀ is the initial radius of the dispersed phase bubbles, while R_M is the largest size of the bubbles. These sizes define M classes, which is named C_i, and the width of the class is given as

\mathrm{\Delta }{R}_i=R_i-R_{i-\mathrm{1}},

while the characteristic size of the class is expressed as

{\overline{R}}_i={\displaystyle \frac{R_{i-\mathrm{1}}+R_i}{\mathrm{2}}}.

The population balance of the bubble sizes between R to R+dR during dt can be expressed as

(1)Q_{\mathrm{i}\mathrm{n}\mathrm{l}\mathrm{e}\mathrm{t}}{\psi }_{\mathrm{i}\mathrm{n}\mathrm{l}\mathrm{e}\mathrm{t}}\mathrm{d}R\mathrm{d}t+r_N\delta (R-R_{\mathrm{0}})V_T\mathrm{d}R\mathrm{d}t+r_{\mathrm{C}}{V}_{\mathrm{T}}\mathrm{d}R\mathrm{d}t=Q_{\mathrm{o}\mathrm{u}\mathrm{t}\mathrm{l}\mathrm{e}\mathrm{t}}\psi \mathrm{d}R\mathrm{d}t+\mathrm{d}(\psi G)+r_{\mathrm{B}}{V}_{\mathrm{T}}\mathrm{d}R\mathrm{d}t+\mathrm{d}(\psi \mathrm{d}R\mathrm{d}t),

where Q_inlet is the inlet volumetric flow rate of the dispersed phase, while Q_outlet is the outlet volumetric flow rate; Ψ_inlet is the number density distribution function of the inlet dispersed phase bubble; Ψ is the number density distribution function; V_T is the total volume of the system, including the dispersed phase and continuous phase; G is the growth rate of the dispersed phase bubbles; r_C is the net rate of creation of the dispersed bubbles due to coalescence in unit volume; r_B is the net rate of disappearance of the dispersed bubbles due to breakage in unit volume; r_N is the net rate of disappearance of the dispersed bubbles due to nucleation in unit volume; and δ is the Dirac Delta function, which is expressed as

(2)\delta (n)=\{\begin{array}{ll}\mathrm{1},& n=\mathrm{0},\\ \mathrm{0},& n\ne \mathrm{0.}\end{array}

By integrating Eq. (1) from R_i-1 to R_i, a new equation can be obtained as

(3){\displaystyle \frac{\mathrm{d}{N}_i}{\mathrm{d}t}}+{\displaystyle \frac{N_i}{V_{\mathrm{T}}}}{\displaystyle \frac{\mathrm{d}{V}_{\mathrm{T}}}{\mathrm{d}t}}+[G(R_i)\psi (R_i)-G(R_{i-\mathrm{1}})\psi (R_{i-\mathrm{1}})]+{\displaystyle \frac{Q_{\mathrm{o}\mathrm{u}\mathrm{t}\mathrm{l}\mathrm{e}\mathrm{t}}{N}_i-Q_{\mathrm{i}\mathrm{n}\mathrm{l}\mathrm{e}\mathrm{t}}{N}_{\mathrm{i},\mathrm{i}\mathrm{n}\mathrm{l}\mathrm{e}\mathrm{t}}}{V_T}}=R_{\mathrm{N},i}+R_{\mathrm{C},i}-R_{\mathrm{B},i},

where R_N,i is the net rate of the creation of the bubbles in the class C_i per unit volume due to nucleation, which is expressed as R_N,i = r_N,i = 1 and R_N,i= 0, i ≠ 1; R_C,i is the net rate of the creation of the bubbles in the class C_i per unit volume due to coalescence; R_B,i is the net rate of the disappearance of the bubbles in the class C_i per unit volume due to breakage; and N_i is the number of the dispersed phase bubbles in the class C_i per unit volume, which is expressed by

(4)N_i(t)={\displaystyle {\int }_{R_{i-\mathrm{1}}}^{R_i}\psi (R,t)\mathrm{d}R}.

Assuming Ψ is constant in the class C_i, which is given as \psi ={\varphi }_i(t), thus

(5)\psi (R_i,t)={\displaystyle \frac{{\varphi }_i+{\varphi }_{i+\mathrm{1}}}{\mathrm{2}}}={\displaystyle \frac{N_i(t)}{\mathrm{2}\mathrm{\Delta }{R}_i}}+{\displaystyle \frac{N_{i+\mathrm{1}}(t)}{\mathrm{2}\mathrm{\Delta }{R}_{i+\mathrm{1}}}},

where

(6)N_i(t)={\varphi }_i(t)\mathrm{\Delta }{R}_i.

Hence Eq. (4) becomes

(7)\{\begin{array}{l}{\displaystyle \frac{\mathrm{d}{N}_{\mathrm{1}}}{\mathrm{d}t}}+{\displaystyle \frac{N_{\mathrm{1}}}{V_{\mathrm{T}}}}{\displaystyle \frac{\mathrm{d}{V}_T}{\mathrm{d}t}}+{\displaystyle \frac{Q_{\mathrm{o}\mathrm{u}\mathrm{t}\mathrm{l}\mathrm{e}\mathrm{t}}{N}_{\mathrm{1}}-Q_{\mathrm{i}\mathrm{n}\mathrm{l}\mathrm{e}\mathrm{t}}{N}_{\mathrm{1},\mathrm{i}\mathrm{n}\mathrm{l}\mathrm{e}\mathrm{t}}}{V_{\mathrm{T}}}}+{\displaystyle \frac{N_{\mathrm{2}}G(R_{\mathrm{1}})}{\mathrm{2}\mathrm{\Delta }{R}_{\mathrm{2}}}}+{\displaystyle \frac{N_{\mathrm{1}}G(R_{\mathrm{1}})}{\mathrm{2}\mathrm{\Delta }{R}_{\mathrm{1}}}}=r_{\mathrm{N}}+R_{\mathrm{C},\mathrm{1}}-R_{\mathrm{B},\mathrm{1}},\hfill \\ {\displaystyle \frac{\mathrm{d}{N}_i}{\mathrm{d}t}}+{\displaystyle \frac{\mathrm{1}}{V_{\mathrm{T}}}}{\displaystyle \frac{\mathrm{d}{V}_{\mathrm{T}}}{\mathrm{d}t}}+{\displaystyle \frac{Q_{\mathrm{o}\mathrm{u}\mathrm{t}\mathrm{l}\mathrm{e}\mathrm{t}}{N}_i-Q_{\mathrm{i}\mathrm{n}\mathrm{l}\mathrm{e}\mathrm{t}}{N}_{\mathrm{1},\mathrm{i}\mathrm{n}\mathrm{l}\mathrm{e}\mathrm{t}}}{V_{\mathrm{T}}}}+{\displaystyle \frac{G(R_i)}{\mathrm{2}\mathrm{\Delta }{R}_{i+\mathrm{1}}}}{N}_{i+\mathrm{1}}+{\displaystyle \frac{G(R_i)-G(R_{i-\mathrm{1}})}{\mathrm{2}\mathrm{\Delta }{R}_i}}{N}_i-{\displaystyle \frac{G(R_{i-\mathrm{1}})}{\mathrm{2}\mathrm{\Delta }{R}_{i-\mathrm{1}}}}{N}_{i-\mathrm{1}}=R_{\mathrm{C},i}-R_{\mathrm{B},i},\hfill \\ {\displaystyle \frac{\mathrm{d}{N}_M}{\mathrm{d}t}}+{\displaystyle \frac{N_M}{V_{\mathrm{T}}}}{\displaystyle \frac{\mathrm{d}{V}_{\mathrm{T}}}{\mathrm{d}t}}+{\displaystyle \frac{Q_{\mathrm{o}\mathrm{u}\mathrm{t}\mathrm{l}\mathrm{e}\mathrm{t}}{N}_M-Q_{\mathrm{i}\mathrm{n}\mathrm{l}\mathrm{e}\mathrm{t}}{N}_{M,\mathrm{i}\mathrm{n}\mathrm{l}\mathrm{e}\mathrm{t}}}{V_{\mathrm{T}}}}-{\displaystyle \frac{G(R_{M-\mathrm{1}})}{\mathrm{2}\mathrm{\Delta }{R}_M}}{N}_M-{\displaystyle \frac{G(R_{M-\mathrm{1}})}{\mathrm{2}\mathrm{\Delta }{R}_{M-\mathrm{1}}}}{N}_{M-\mathrm{1}}=R_{\mathrm{C},M}-R_{\mathrm{B},M}.\hfill \end{array}

Auxiliary equations

The growth rate of the dispersed phase bubbles can be derived from mass and energy balance, which are shown as

(8)\{\begin{array}{l}{\rho }_{\mathrm{D}\mathrm{1}}\mathrm{d}{V}_{\mathrm{D}\mathrm{1}}={\rho }_{\mathrm{D}\mathrm{g}}\mathrm{d}{V}_{\mathrm{D}\mathrm{g}},\hfill \\ \mathrm{d}{V}_{\mathrm{D}\mathrm{g}}-\mathrm{d}{V}_{\mathrm{D}\mathrm{1}}=\mathrm{d}\left({\displaystyle \frac{\mathrm{4}\mathrm{\pi }}{\mathrm{3}}}{R}^{\mathrm{3}}\right),\hfill \\ \mathrm{\pi }{D}^{\mathrm{2}}{h}_{\mathrm{f}}(T_{\mathrm{C}}-T_{\mathrm{D}\mathrm{S}})\mathrm{d}t=L_{\mathrm{D}}{\rho }_{\mathrm{D}\mathrm{1}}\mathrm{d}{V}_{\mathrm{D}\mathrm{1}}.\hfill \end{array}

From Eq. (8), the growth rate of the dispersed phase bubbles can be expressed as

(9)G(R)={\displaystyle \frac{\mathrm{d}R}{\mathrm{d}t}}={\displaystyle \frac{h_{\mathrm{f}}(T_{\mathrm{C}}-T_{\mathrm{D}\mathrm{s}})({\rho }_{\mathrm{D}\mathrm{l}}-{\rho }_{\mathrm{D}\mathrm{v}})}{{\rho }_{\mathrm{D}\mathrm{l}}{\rho }_{\mathrm{D}\mathrm{g}}{L}_{\mathrm{D}}}},

where T_C is the temperature of the continuous phase liquid; T_Ds is the temperature of the dispersed phase, which is assumed to be the temperature of the saturated temperature of the dispersed phase and remains constant in this case; ρ_Dl is the density of the dispersed phase liquid; ρ_Dv is the density of the dispersed phase vapor; L_D is the latent heat of vaporization of the dispersed phase; and h_f is the heat transfer coefficient for a single dispersed bubble.

The correlation of Nusselt number of the single droplet proposed by Battya et al. [22] which has been confirmed to be applicable to any immiscible dispersed and continuous phase, can be expressed as

(10)Nu=\mathrm{0.64}J{a}^{-\mathrm{0.35}}P{e}^{\mathrm{0.5}},

where Ja is the Jakob number, which is expressed as

Ja={\displaystyle \frac{{\rho }_{\mathrm{C}}{C}_{\mathrm{p}\mathrm{C}}(T_{\mathrm{C}}-T_{\mathrm{D}\mathrm{s}})}{{\rho }_{\mathrm{D}\mathrm{v}}{L}_{\mathrm{D}}}};

Pe is the Peclet number, which is expressed as

Pe={\displaystyle \frac{\mathrm{2}RU}{{\alpha }_{\mathrm{C}}}};

C_pC is the specific heat of the continuous phase liquid; α_C is the thermal diffusivity of the continuous phase liquid; and U is the instantaneous velocity of the single rising bubble. Thus the correlation of the heat transfer coefficient for a single dispersed bubble is derived as

(11)h_{\mathrm{f}}={\displaystyle \frac{\mathrm{0.32}{k}_{\mathrm{C}}}{R}}{\left({\displaystyle \frac{\mathrm{2}UR}{{\alpha }_{\mathrm{C}}}}\right)}^{\mathrm{0.5}}{\left[{\displaystyle \frac{{\rho }_{\mathrm{C}}{C}_{\mathrm{p}\mathrm{C}}(T_{\mathrm{C}}-T_{\mathrm{D}\mathrm{s}})}{{\rho }_{\mathrm{D}\mathrm{v}}{L}_{\mathrm{D}}}}\right]}^{\mathrm{0.35}},

where k_C is the thermal conductivity of the continuous phase liquid.

The correlation of the instantaneous velocity of the single rising bubble is given by Raina et al. [21] as

(12)U=\mathrm{1.91}{\left\{\left[\mathrm{1}-{\displaystyle \frac{{\rho }_{\mathrm{D}\mathrm{v}}}{{\rho }_{\mathrm{C}}}}{\left({\displaystyle \frac{R_{\mathrm{0}}}{R}}\right)}^{\mathrm{3}}\right]{\displaystyle \frac{{\sigma }_{\mathrm{C}}}{\mathrm{2}R{\rho }_{\mathrm{C}}}}\right\}}^{\mathrm{1}/\mathrm{2}}{\displaystyle \frac{{(\mathrm{1}-ϵ)}^{\mathrm{2}}}{\mathrm{1}-{ϵ}^{\mathrm{5}/\mathrm{3}}}},

where R₀ is the initial radius of the dispersed phase droplet; σ_C is the interfacial tension of the continuous phase liquid; and ϵ is the dispersed phase holdup.

As the distribution of the dispersed phase bubble size is obtained by the population balance model, the number of the bubbles of each class can be determined. Hence, the overall heat transfer rate Q can be calculated by

(13)Q={\displaystyle \sum _{i=\mathrm{1}}^N{Q}_i}=\mathrm{4}\mathrm{\pi }{\displaystyle \sum _{i=\mathrm{1}}^N{R}_i^{\mathrm{2}}{N}_i(T_{\mathrm{C}}-T_{\mathrm{D}\mathrm{s}})h_{\mathrm{f}}}.

Assuming the temperature of the dispersed phase keeps constant during the evaporating rising process, the continuously changing temperature of the continuous phase liquid can be calculated from the energy balance

(14)T_{\mathrm{C}}(t+\mathrm{\Delta }t)={\displaystyle \frac{Q_{\mathrm{i}\mathrm{n}\mathrm{l}\mathrm{e}\mathrm{t}}{T}_{\mathrm{D}\mathrm{s}}+V_{\mathrm{T}}(t)T_{\mathrm{C}}(t)/\mathrm{\Delta }t-Q/({\rho }_{\mathrm{C}}{C}_{\mathrm{p}\mathrm{C}})}{V_{\mathrm{T}}(t+\mathrm{\Delta }t)/\mathrm{\Delta }t+Q_{\mathrm{o}\mathrm{u}\mathrm{t}\mathrm{l}\mathrm{e}\mathrm{t}}}}.

The total volume of the system V_T during the process can be determined as

(15)V_{\mathrm{T}}(t+\mathrm{\Delta }t)={\displaystyle \frac{V}{\mathrm{1}-{\displaystyle \frac{\mathrm{4}}{\mathrm{3}}}\mathrm{\pi }{\displaystyle \sum _{i=\mathrm{1}}^N{\overline{R}}_i^{\mathrm{3}}{N}_i}}},

where V is the volume of the continuous phase liquid, which is considered to be constant in this case. Then the changing rate of the volume is determined by

(16){\displaystyle \frac{\mathrm{d}{V}_{\mathrm{T}}(t+\mathrm{\Delta }t)}{\mathrm{d}t}}={\displaystyle \frac{V_{\mathrm{T}}(t)-V_{\mathrm{T}}(t-\mathrm{\Delta }t)}{\mathrm{\Delta }t}}.

Finally, the volumetric heat transfer coefficient U_{\mathrm{v}} is defined as

(17)U_{\mathrm{v}}={\displaystyle \frac{Q}{V_{\mathrm{T}}(T_{\mathrm{C}}-T_{\mathrm{D}\mathrm{s}})}}.

Results and discussion

To solve Eqs. (12)–(17), a series of simplification and assumption are conducted in advance. First, as the dispersed phase flow injects into the continuous phase liquid gently, and the density of the disperse phase bubbles is low, the influences of the coalescence and breakage are neglected in this case. Considering that the continuous phase liquid is water, which is not frozen to be ice during the heat transfer process, and the dispersed phase is butane, the effect of the nucleation is also ignored. In this case, Eq. (7) can be reduced to be homogeneous equations. Assuming the volume of the continuous phase liquid is set at V = 0.1 m³; the initial temperature of the continuous phase liquid is 277 K, while the temperature of the dispersed phase keeps at saturated temperature 272 K; In the calculation, 100 droplets of the dispersed phase liquid are injected into the continuous phase liquid, and the injecting rate of the dispersed phase liquid is 100 per second. The fourth Runga-Kutta method is used in the current calculation.

The heat transfer coefficient h_f for a single dispersed bubble with different initial size decreases slowly as the bubble grows, as demonstrated in Fig. 2. This variation confirms the point of view that most heat transfer occurs on the interface between un-evaporated dispersed phase and the continuous phase. As the bubble evaporates during the rising process, this area decreases obviously.

The distribution of the number of the dispersed phase bubbles in the system at different time is displayed in Fig. 3. As the dispersed phase droplets inject into the continuous phase, the total number of the dispersed bubbles becomes increasing rapidly. As the time goes, the peak of the number of the dispersed phase bubbles moves to the larger value of R/R₀, indicating the growing of the bubble size. This trend is remarkable at the beginning.

Figure 4 shows that the temperature of the continuous phase liquid decreases with the time. The temperature decreases more rapidly with larger initial radius of the dispersed bubble.

The variations of the total heat transfer rate with time are depicted in Fig. 5. It can be observed that the total heat transfer rate increases first and then decreases. The reason for is that, at the beginning the volume of the dispersed phase, which is injected into the continuous phase liquid, is the main influencing factor, and this effect nevertheless decreases when the temperature of the continuous phase liquid reduces.

It can be noticed from Fig. 6 that the volumetric heat transfer coefficient increases with the time. Moreover, as the initial radius of the dispersed phase droplet gets larger, the volumetric heat transfer coefficient increases faster.

The influence of the different initial temperature of the continuous phase liquid on the volumetric heat transfer coefficient is presented in Fig. 7. It can be observed that the effect of the different initial temperature of the continuous phase is small enough to be neglected.

Due to the lack of experimental data of the butane (R600)/water system, the experimental data for the volumetric heat transfer coefficient of R114/water system obtained in a column of 0.114 m diameter [33], are selected for the comparison with the calculation results in this paper. It can be seen from Fig. 8 that when the flow rate of the dispersed phase is set at 140 kg/h, the calculation results are in reasonable agreement with the experimental data, indicating that the model proposed in this paper is suitable for the prediction of the direct contact heat transfer between the refrigerant and water.

Conclusions

For the purpose of investigating the seawater desalination system utilizing LNG cold energy, the heat transfer mechanism of this system was conducted. The physical model and mathematical model, including the population balance equation and other auxiliary equations, were proposed and analyzed. It is found that the peak of the number of the dispersed phase bubbles moves to the larger value of R/R₀ and this trend is remarkable at the beginning. The temperature of the continuous phase liquid decreases with time, and the temperature decreases rapidly with larger initial radius of the dispersed bubble. The total heat transfer rate increases first and then decreases, due to different influencing factors. The volumetric heat transfer coefficient increases with time, and its increasing rate is higher at larger initial radius of the dispersed phase droplet. In addition, the influence of the different initial temperature of the continuous phase liquid is small and can be neglected. The calculated results of the volumetric heat transfer coefficient are in reasonable agreement with the experimental data of R114/water system.

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