3. The model applicable to the proposed GALHP

The mathematical model for the evaporation-condensation processes in the proposed GALHP is based on the following assumptions:


Study of a Novel Liquid-Vapour Separator-Incorporated Gravitational Loop Heat Pipe DOI: http://dx.doi.org/10.5772/intechopen.86048


The mass flow rate within the wick structure is considered to be constant owing to the mass conservation law, given by [4, 16]

$$
\dot{m} = \rho\_l \mu\_l 2\pi \chi L\_{ewa} \varepsilon\_w = \frac{\dot{Q}}{h\_{\text{fg}}} \tag{1}
$$

#### 3.1 Energy conservation and temperature profile in the evaporator

The energy conservation equations of the single wick structure are given by [4, 25]

$$\frac{\dot{m}\,\mathrm{C}\_{pl}}{2\pi r L\_{\mathrm{eva}}} \frac{\partial T\_w}{\partial r} = k\_{\mathrm{eff}} \frac{1}{r} \frac{\partial}{\partial r} \left(\mathrm{r}\frac{\partial \mathrm{T}\_w}{\partial r}\right) \tag{2}$$

The effective thermal conductivity of liquid-saturated wick in a cylindrical geometry is [3, 4]

$$k\_{\epsilon\overline{f}} = \frac{k\_l[(k\_l + k\_w) - (\mathbf{1} - \varepsilon\_w)(k\_l - k\_w)]}{[(k\_l + k\_w) - (\mathbf{1} - \varepsilon\_w)(k\_l - k\_w)]} \tag{3}$$

in which, the porosity of screen wick is expressed as [3, 4]

$$
\varepsilon\_w = 1 - \frac{1.05\pi n\_w D\_w}{4} \tag{4}
$$

Define the variable α as

$$\alpha = \frac{\dot{m}C\_{pl}}{2\pi r L\_{eva} k\_{eff}} \tag{5}$$

Then, rewrite Eq. (2) as

$$\frac{\partial^2 T\_w}{\partial r^2} + \frac{1}{r}(\mathbf{1} - a)\frac{\partial \Gamma\_\mathbf{w}}{\partial r} = \mathbf{0} \tag{6}$$

The boundary conditions are

$$\begin{cases} \left. T(r) \right|\_{r=r\_{\nu,0}} = \left. T\_{w,0} \right| \\\\ \left. T(r) \right|\_{r=r\_{\nu,i}} = \left. T\_{w,i} \right| \end{cases} \tag{7}$$

will flow upward through the central tubular pipe and enter into the vapour transport line. If the liquid level is further controlled by a valve mounted on the liquid transfer line, the downward liquid flow rate can be controlled to match the rate of evaporation on the inner surface of the heat pipe. So the wicked inner heat pipe surface will be constantly in 'wet'state, thus preventing the potential 'dry-out' problem with the conventional GALHP. Meanwhile, the vapour and liquid flows will be regulated in the same direction and separated clearly during the operation, thus preventing the potential entrainment problem with the conventional straight

Based on the above innovative concept, a dedicated mathematical model and the associated computer program will be developed to analyse the characteristics of the

The mathematical model for the evaporation-condensation processes in the pro-

• Heat transfer and fluid flow occur under the quasi-steady-state condition.

• Heat conduction and fluid flow across the wick are one-dimensional in the

• Heat pipe evaporator is heated axial-symmetrically and the difference of the

• The hydrostatic pressure drop across the radial direction owing to the gravity

3. The model applicable to the proposed GALHP

Top-positioned vapour-liquid separator of new GALHP.

Recent Advances in Heat Pipes

posed GALHP is based on the following assumptions:

temperature along the axial direction is negligible.

heat pipe.

Figure 2.

new GALHP.

radial direction.

52

effect is considered to be zero.

By solving the second-order ordinary differential Eq. (6) with twice integrals, temperature distribution in the wick is

$$T(r) = \left[\frac{T\_{w,0} - T\_{w,i}}{\left(r\_{w,0} - r\_{w,i}\right)^{\alpha} - 1}\right] \times \left(\frac{r}{r\_{w,i}}\right)^{\alpha} - \frac{T\_{w,0} - T\_{w,i}(r\_{w,0} - r\_{w,i})^{\alpha}}{\left(r\_{w,0} - r\_{w,i}\right)^{\alpha} - 1} \tag{8}$$

For a single saturated wick layer, the radial thermal conductance is then

$$G\_w = \frac{k\_{\text{eff}} A\_{w,i}}{T\_{w,0} - T\_{w,i}} \times \frac{\partial T\_w}{\partial r} \bigg|\_{r=r\_{w,i}} = \frac{\dot{m} \, \mathcal{C}\_{pl}}{(r\_{w,0} - r\_{w,i})^a - 1} \tag{9}$$

keff,<sup>0</sup>

DOI: http://dx.doi.org/10.5772/intechopen.86048

Putting Eq. (10) into the above expression results in

Tw, <sup>1</sup> � Tw, <sup>2</sup>

The thermal resistance in this region is therefore

Reva ¼

4. Design and fabrication of the proposed GALHP

Put Eq. (11) into Eq. (16) and it becomes

Rewrite Eq. (17) as

3.2 Flow characteristic

condenser and liquid line.

55

structure,

∂Tw ∂r <sup>r</sup>¼<sup>r</sup> w, <sup>2</sup><sup>þ</sup>

 

Study of a Novel Liquid-Vapour Separator-Incorporated Gravitational Loop Heat Pipe

ð Þ <sup>r</sup>1=r<sup>2</sup> <sup>α</sup><sup>1</sup> � <sup>1</sup> <sup>¼</sup> Tw, <sup>2</sup> � Tw, <sup>3</sup>

¼ keff,i

ð Þ <sup>r</sup>2=r<sup>3</sup> <sup>α</sup><sup>2</sup> � <sup>1</sup> 

Gw,0ð Þ¼ Tw, <sup>1</sup> � Tw,<sup>2</sup> Gw,ið Þþ Tw, <sup>2</sup> � Tw,<sup>3</sup> mC\_ plð Þ Tw, <sup>2</sup> � Tw, <sup>3</sup> (17)

Tw, <sup>2</sup> <sup>¼</sup> Gw,0Tw, <sup>1</sup> <sup>þ</sup> Gw,i <sup>þ</sup> mC\_ pl Tw, <sup>3</sup> Gw,<sup>0</sup> þ Gw,i þ mC\_ pl

Put Eq. (18) into Eq. (14) for obtaining the expression of the composite wick

ln <sup>r</sup><sup>0</sup> r1 

2πLevakeva,wall

The interface temperature conditions can be assumed local thermal equilibrium

In a heat pipe, the maximum capillary pumping head (ΔPc,max) must be greater than or at least equal to the total pressure drops (ΔP) along the heat pipe. The total pressure drops (ΔP) should be the sum of pressure drops in all the heat pipe components, that is, wick structure, evaporator, three-way separator, vapour line,

Based on the results derived from the theoretical and computer simulation studies [26], the proposed GALHP was designed, fabricated and presented in Figure 3 respectively. For the evaporator, the length remained 550 mm and diameter fixed to 22 mm. Within the inner surfaces of the evaporator, the

compound screen mesh wick structure was applied with the size of 160 � 60 mm.

þ 1 Gw

Tw,<sup>3</sup> ¼ Tv ¼ Tint (21)

ΔPc,max þ ΔPg ≥ΔP (22)

ΔP ¼ ΔPw þ ΔPeva þ ΔPtw þ ΔPvl þ ΔPcond þ ΔPll (23)

Gw,<sup>0</sup> þ Gw,i þ mC\_ pl

Gw <sup>¼</sup> Gw,0Gw,i

∂Tw ∂r

 r¼rw, <sup>2</sup>�

� r2 r3 <sup>α</sup><sup>2</sup> (15)

(16)

(18)

(19)

(20)

In this case, the temperature distributions are

$$\begin{cases} r\_3 \le r \le r\_2, \; T(r) = \left[ \frac{T\_{w,2} - T\_{w,3}}{(r\_2/r\_3)^{\alpha\_1} - 1} \right] \times \left( \frac{r}{r\_3} \right)^{\alpha\_1} - \frac{T\_{w,2} - T\_{w,3}(r\_2/r\_3)^{\alpha\_2}}{(r\_2/r\_3)^{\alpha\_1} - 1} \\\\ r\_2 \le r \le r\_1, \; T(r) = \left[ \frac{T\_{w,1} - T\_{w,2}}{(r\_1/r\_2)^{\alpha\_1} - 1} \right] \times \left( \frac{r}{r\_2} \right)^{\alpha\_1} - \frac{T\_{w,1} - T\_{w,2}(r\_1/r\_2)^{\alpha\_1}}{(r\_1/r\_2)^{\alpha\_1} - 1} \\\\ r\_1 \le r \le r\_0, \; T(r) = T\_{w,1} + \frac{(T\_{env,wall} - T\_{w,1})}{\ln \left( \frac{r\_0}{r\_1} \right)} \ln \left( \frac{r}{r\_1} \right) \end{cases} \tag{10}$$

Thermal conductance of the inner and outer wick layers are respectively

$$\begin{cases} G\_{w,0} = \frac{k\_{\text{eff},0} A\_{w,2}}{T\_{w,1} - T\_{w,2}} \times \frac{\partial T\_w}{\partial r} \bigg|\_{r=r\_{w,2}} = \frac{\dot{m} \mathbf{C}\_{pl}}{(r\_1/r\_2)^{\alpha\_1} - \mathbf{1}} \\\\ G\_{w,i} = \frac{k\_{\text{eff},i} A\_{w,3}}{T\_{w,2} - T\_{w,3}} \times \frac{\partial T\_w}{\partial r} \bigg|\_{r=r\_{w,3}} = \frac{\dot{m} \mathbf{C}\_{pl}}{(r\_2/r\_3)^{\alpha\_2} - \mathbf{1}} \end{cases} \tag{11}$$

and

$$\begin{cases} \begin{aligned} \alpha\_{1} &= \frac{\dot{m}C\_{pl}}{2\pi k\_{\text{eff},0}L\_{\text{emu}}} \\ \alpha\_{2} &= \frac{\dot{m}C\_{pl}}{2\pi k\_{\text{eff},i}L\_{\text{emu}}} \end{aligned} \end{cases} \tag{12}$$

$$\begin{cases} k\_{\text{eff},0} = \frac{k\_{l}[(k\_{l}+k\_{w,0})-(1-\varepsilon\_{w,0})(k\_{l}-k\_{w,0})]}{[(k\_{l}+k\_{w,0})-(1-\varepsilon\_{w,0})(k\_{l}-k\_{w,0})]} \\\\ k\_{\text{eff},i} = \frac{k\_{l}[(k\_{l}+k\_{w,i})-(1-\varepsilon\_{w,i})(k\_{l}-k\_{w,i})]}{[(k\_{l}+k\_{w,i})-(1-\varepsilon\_{w,i})(k\_{l}-k\_{w,i})]} \end{aligned} \tag{13}$$

As a result, the overall thermal conductance of the composite wick structure can be given by

$$\mathbf{G}\_{w} = \frac{k\_{\rm eff,i} A\_{w,3}}{T\_{w,1} - T\_{w,3}} \times \left. \frac{\partial T\_{w}}{\partial r} \right|\_{r=u\_{3}} = \frac{T\_{w,2} - T\_{w,3}}{T\_{w,1} - T\_{w,3}} \times \left. \frac{k\_{\rm eff,i} A\_{w,3}}{T\_{w,2} - T\_{w,3}} \times \left. \frac{\partial T\_{w}}{\partial r} \right|\_{r=r\_{w,1}} = \frac{T\_{w,2} - T\_{w,3}}{T\_{w,1} - T\_{w,3}} \times G\_{w,1} \tag{14}$$

According to energy conservation, heat flux at the internal surface of the outer wick layer should be equal to the heat flux at the external surface of the inner wick layer

Study of a Novel Liquid-Vapour Separator-Incorporated Gravitational Loop Heat Pipe DOI: http://dx.doi.org/10.5772/intechopen.86048

$$k\_{\sharp\overline{f},0} \frac{\partial T\_w}{\partial r}\bigg|\_{r=r\_{w,2^+}} = k\_{\sharp\overline{f},i} \frac{\partial T\_w}{\partial r}\bigg|\_{r=r\_{w,2^-}}\tag{15}$$

Putting Eq. (10) into the above expression results in

$$\frac{T\_{w,1} - T\_{w,2}}{(r\_1/r\_2)^{a\_1} - \mathbf{1}} = \left[\frac{T\_{w,2} - T\_{w,3}}{(r\_2/r\_3)^{a\_2} - \mathbf{1}}\right] \times \left(\frac{r\_2}{r\_3}\right)^{a\_2} \tag{16}$$

Put Eq. (11) into Eq. (16) and it becomes

$$G\_{w,0}(T\_{w,1} - T\_{w,2}) = G\_{w,i}(T\_{w,2} - T\_{w,3}) + \dot{m}C\_{pl}(T\_{w,2} - T\_{w,3})\tag{17}$$

Rewrite Eq. (17) as

By solving the second-order ordinary differential Eq. (6) with twice integrals,

� Tw,<sup>0</sup> � Tw,ið Þ rw,<sup>0</sup> � rw,i <sup>α</sup>

<sup>¼</sup> mC\_ pl

ð Þ rw,<sup>0</sup> � rw,i <sup>α</sup> � <sup>1</sup> (8)

ð Þ rw,<sup>0</sup> � rw,i <sup>α</sup> � <sup>1</sup> (9)

� Tw, <sup>2</sup> � Tw, <sup>3</sup>ð Þ <sup>r</sup>2=r<sup>3</sup> <sup>α</sup><sup>2</sup> ð Þ <sup>r</sup>2=r<sup>3</sup> <sup>α</sup><sup>2</sup> � <sup>1</sup>

� Tw, <sup>1</sup> � Tw,2ð Þ <sup>r</sup>1=r<sup>2</sup> <sup>α</sup><sup>1</sup> ð Þ <sup>r</sup>1=r<sup>2</sup> <sup>α</sup><sup>1</sup> � <sup>1</sup>

� � ln <sup>r</sup>

r1 � � (10)

(11)

(12)

(13)

� Gw,i

(14)

ln <sup>r</sup><sup>0</sup> r1

<sup>¼</sup> mC\_ pl ð Þ <sup>r</sup>1=r<sup>2</sup> <sup>α</sup><sup>1</sup> � <sup>1</sup>

<sup>¼</sup> mC\_ pl ð Þ <sup>r</sup>2=r<sup>3</sup> <sup>α</sup><sup>2</sup> � <sup>1</sup>

r rw,i � �<sup>α</sup>

For a single saturated wick layer, the radial thermal conductance is then

� r r3 � �<sup>α</sup><sup>2</sup>

� r r2 � �<sup>α</sup><sup>1</sup>

Thermal conductance of the inner and outer wick layers are respectively

� ∂Tw ∂r � � � � r¼rw, <sup>2</sup>

� ∂Tw ∂r � � � � r¼rw, <sup>3</sup>

<sup>α</sup><sup>1</sup> <sup>¼</sup> mC\_ pl 2πkeff,0Leva

<sup>α</sup><sup>2</sup> <sup>¼</sup> mC\_ pl 2πkeff,iLeva

keff,<sup>0</sup> <sup>¼</sup> kl½ � ð Þ� kl <sup>þ</sup> kw,<sup>0</sup> ð Þ <sup>1</sup> � <sup>ε</sup>w,<sup>0</sup> ð Þ kl � kw,<sup>0</sup> ½ � ð Þ� kl þ kw,<sup>0</sup> ð Þ 1 � εw,<sup>0</sup> ð Þ kl � kw,<sup>0</sup>

keff,i <sup>¼</sup> kl ð Þ� kl <sup>þ</sup> kw,i ð Þ <sup>1</sup> � <sup>ε</sup>w,i ð Þ kl � kw,i ½ � ð Þ� kl þ kw,i ð Þ 1 � εw,i ð Þ kl � kw,i ½ �

As a result, the overall thermal conductance of the composite wick structure can

According to energy conservation, heat flux at the internal surface of the outer wick layer should be equal to the heat flux at the external surface of

� keff,iAw,<sup>3</sup> Tw, <sup>2</sup> � Tw, <sup>3</sup> � <sup>∂</sup>Tw ∂r

� � � � r¼rw, <sup>3</sup>

<sup>¼</sup> Tw, <sup>2</sup> � Tw, <sup>3</sup> Tw, <sup>1</sup> � Tw, <sup>3</sup>

<sup>r</sup><sup>1</sup> <sup>≤</sup>r≤r0,T rð Þ¼ Tw, <sup>1</sup> <sup>þ</sup> ð Þ Teva,wall � Tw, <sup>1</sup>

�

� ∂Tw ∂r � � � � r¼rw,i

temperature distribution in the wick is

ð Þ rw,<sup>0</sup> � rw,i <sup>α</sup> � <sup>1</sup> � �

Gw <sup>¼</sup> keffAw,i

In this case, the temperature distributions are

<sup>r</sup><sup>3</sup> <sup>≤</sup> <sup>r</sup><sup>≤</sup> <sup>r</sup>2,T rð Þ¼ Tw, <sup>2</sup> � Tw,<sup>3</sup>

8

>>>>>>>>>>><

>>>>>>>>>>>:

and

be given by

Gw <sup>¼</sup> keff,iAw, <sup>3</sup> Tw, <sup>1</sup> � Tw,<sup>3</sup>

the inner wick layer

54

<sup>r</sup><sup>2</sup> <sup>≤</sup>r≤r1,T rð Þ¼ Tw, <sup>1</sup> � Tw, <sup>2</sup>

8 >>>><

>>>>:

8 >>><

>>>:

� <sup>∂</sup>Tw ∂r <sup>r</sup>¼rw, <sup>3</sup>

� � � � Tw,<sup>0</sup> � Tw,i

ð Þ <sup>r</sup>2=r<sup>3</sup> <sup>α</sup><sup>2</sup> � <sup>1</sup> � �

ð Þ <sup>r</sup>1=r<sup>2</sup> <sup>α</sup><sup>1</sup> � <sup>1</sup> � �

Gw,<sup>0</sup> <sup>¼</sup> keff,0Aw,<sup>2</sup>

Gw,i <sup>¼</sup> keff,iAw, <sup>3</sup>

Tw, <sup>1</sup> � Tw, <sup>2</sup>

Tw, <sup>2</sup> � Tw, <sup>3</sup>

8 >>><

>>>:

<sup>¼</sup> Tw, <sup>2</sup> � Tw, <sup>3</sup> Tw, <sup>1</sup> � Tw, <sup>3</sup>

T rð Þ¼ Tw,<sup>0</sup> � Tw,i

Recent Advances in Heat Pipes

$$T\_{w,2} = \frac{G\_{w,0}T\_{w,1} + \left(G\_{w,i} + \dot{m}C\_{pl}\right)T\_{w,3}}{G\_{w,0} + G\_{w,i} + \dot{m}C\_{pl}}\tag{18}$$

Put Eq. (18) into Eq. (14) for obtaining the expression of the composite wick structure,

$$G\_w = \frac{G\_{w,0} G\_{w,i}}{G\_{w,0} + G\_{w,i} + \dot{m} C\_{pl}} \tag{19}$$

The thermal resistance in this region is therefore

$$R\_{\rm eva} = \frac{\ln\left(\frac{r\_0}{r\_1}\right)}{2\pi L\_{\rm eva} k\_{\rm eva, wall}} + \frac{1}{G\_w} \tag{20}$$

The interface temperature conditions can be assumed local thermal equilibrium

$$T\_{w,3} = T\_v = T\_{\text{int}} \tag{21}$$

#### 3.2 Flow characteristic

In a heat pipe, the maximum capillary pumping head (ΔPc,max) must be greater than or at least equal to the total pressure drops (ΔP) along the heat pipe. The total pressure drops (ΔP) should be the sum of pressure drops in all the heat pipe components, that is, wick structure, evaporator, three-way separator, vapour line, condenser and liquid line.

$$
\Delta P\_{c,\max} + \Delta P\_{\mathcal{g}} \ge \Delta \mathcal{P} \tag{22}
$$

$$
\Delta \mathbf{P} = \Delta P\_w + \Delta P\_{em} + \Delta P\_{tw} + \Delta P\_{vl} + \Delta P\_{cond} + \Delta P\_{ll} \tag{23}
$$

#### 4. Design and fabrication of the proposed GALHP

Based on the results derived from the theoretical and computer simulation studies [26], the proposed GALHP was designed, fabricated and presented in Figure 3 respectively. For the evaporator, the length remained 550 mm and diameter fixed to 22 mm. Within the inner surfaces of the evaporator, the compound screen mesh wick structure was applied with the size of 160 � 60 mm. For the condenser, they were all fixed with a steel cooling jacket of the same size, with a length of 150 mm and a diameter of 105 mm. The detailed design parameters are illustrated in Table 1.

5. Experimental set-up and procedure

DOI: http://dx.doi.org/10.5772/intechopen.86048

Figure 4. On-site testing rig.

57

5.1 Experimental set-up and instrumentation

Figure 4 shows the test rig of the proposed GALHP. In the rig, an electrical heating tap with the percentage controller, which acts as the heat source, was evenly attached to the external surfaces of the evaporators. The condenser is covered by a steel cooling jacket that allows cooling water to pass through, removing heat from the condenser. A magnetic regeneration water pump was installed in the cooling water loop to power the cooling water cross. A clamp-supported retort stand was used to adjust the inclination angle of the piping installation. The foamy polyurethane was attached to the pipes to provide a satisfactory insulation. During operation, in order to keep a relatively constant condensation temperature, the water tap would remain open to enable adequate amount of cold water to be fed into the loop. When the water tank was fully charged, the drainage valve would be

Study of a Novel Liquid-Vapour Separator-Incorporated Gravitational Loop Heat Pipe

A list of the piping elements and test instruments are provided in Table 2. A number of T-type thermocouples were attached to the external surface of heat pipe walls, and installed in the inlet/outlet and inside of cooling jacket and water tank: there were totally four thermocouples (No. 1–4) equidistantly attached along each heat pipe evaporator wall from top to bottom, which were used to measure the temperature distribution along the evaporator wall and their corresponding average

temperature at the evaporation sections; another four thermocouples were

respectively placed in the mid of heat pipe condenser wall (No. 7), the inlet/outlet

turned open to allow the extra amount of water to be discharged.

Figure 3. Fabrication schematics of the proposed GALHP.


#### Table 1.

Design parameters of the proposed GALHP.

Study of a Novel Liquid-Vapour Separator-Incorporated Gravitational Loop Heat Pipe DOI: http://dx.doi.org/10.5772/intechopen.86048
