Description of the model

Shafii and Faghri [21] developed a model ofthe PHP where fluid oscillations occur as a result of heat transfer. The mass, momenturn and energy equations are solved in a 1-dimensional form.

The system, consisting of liquid slugs and vapour plugs is modelled as depicted in tigure 3-3.

D

Figure 3-3 One-dimensional model of the pulsating heat pipe

3.2.1.1 Momenturn equation

Several forces act upon the liquid slugs, which can cause the slug to move. These forces are:

• force due to pressure difference of the adjoining vapour plugs

• frictional force

• gravitational force

• capillary force

Force due to pressure difference ofthe adjoining vapour plugs

The vapour plugs on either side of the liquid slug can be at different pressures, due to heating. The force,

Fp

(N) acting upon the liquid slug i is expressed as:

Fp,i

=

(P;,,i - P;,,i+l) ·A ^(3.4) where Pv,ds the pressure ofvapourplug i (N/m²), Pvi+I pressure ofthe vapour plug i+l (N/m²) and A the cross-sectional area ofthe tube (m2).

Frictional force

As the liquid slugs moves through the tube a frictional force will exist due to shear stress.

The shear stress 't (N/m²) is expressed as:

r = 0.5 ·

f ·

p1 •

vi

^(3.5)

where fis the friction coefficient (-) , PI the density of the liquid (kg!m\ and VI the velocity of the liquid slug (m/s).

The friction coefficient is determined as [21]:

f =}!

ifRe

~

¹¹⁸⁰

Re

f

= 0.078Re-0.² ifRe > 1180

The friction force on liquid slug i acts opposite to the direction of the velocity and is calculated as:

(3.6) (3.7)

F.=tr·d·L.·r _{J,z l,z} (3.8)

where dis the diameter ofthe tube (m) and L1,i is the length ofliquid slug (m).

Gravitational force

Since the liquid slugs are located in the bottorn turns the gravitational force acting on the liquid slug depends on the height difference ofthe water columns (figure 3-4). The gravitational force

Fg

(N) on liquid slug i is given by:

Fg,i

=

p1 • g ·A· dh1,i (3.9) where g is the gravitational constant (m/s²⁾ and

dht,i

is the difference in height of the water columns ofliquid slug i (m).

Figure 3-4 Definition of the height difference dh

Capillary force

The capillary pressure drop across the liquid vapour interface 8P(N/m²) is related to the surface tension cr (N/m) and the principle radii r1 and rn (m):

!::.p

= u.(_!_+_!_)

^{(3.1 0)}

ri ru

Assuming that the liquid vapour interface is spherical at both the receding and advancing end of the plug, the radii r 1 = r1 = rn and r2 = r1 = rn for the receding and advancing end respectively, see tigure 3-5. The radius of curvature at the receding and advancing end are calculated by:

d d

r1

= ;

^r2

=

2·cos(Br) 2·cos(BJ (3.11)

where dis the diameter ofthe tube (m) ander and ea the receding and advancing contact angle (rad), see tigure 3-5.

lf the two contact angles of the slug differ the pressure difference across the plug due to capillary pressure can he calculated by substituting equation 3.11 in 3.10:

l::.pcap =4·u(cos0r -cosBa)/d (3.12)

\./

Figure 3-5 Advancing (Ba) and receding (Br) contact angles ofliquid slug

The values of ()r and ()a depends on the velocity with which the liquid moves over the surface [22]. A typical variation ofthe advancing and receding contact angle with velocity is depicted in tigure 3-6.

The static contact angles are dependent on the history of the movement This

phenomenon is called contact angle hysteresis, which is caused by surface roughness and contamination of the surface [23]. It is also dependent on the combination of liquid and solid surface. For instanee the combination of water and aluminium is reported to have a static advancing contact angle of 73° and a receding contact angle of 34°[24].

However no exact relations exist for the contact angles as a function ofvelocity.

Therefore an estimation should be done.

In general the difference between advancing and receding contact angle increases with increasing velocity. The maximum pressure difference across a liquid plug exist when Sa is 90° and Sr is 0°. It is therefore assumed that at maximum liquid plug velocity V1max

(m/s) the pressure difference is maximum (.ó.Pmax (N/m²)). The resulting pressure difference is then expressed as:

M _cap =V·~ _I M

81-1

0

v/max

Figure 3-6 Contact angles as a function of liquid slug velocity

(3.13)

V

Combining the contributions of all forces results in the following form ofthe momenturn equation:

dm₁.v₁^.

_.:..:.: ·'---'-"- · ' = (P"

^{i -}

P"

^i+I )A -1ldL₁ :r + p₁gAdh-~an ·A

dl ' ' '¹ "

(3.14) where mt,i is the mass ofliquid slug i and Vt,i the velocity ofliquid slug i.

3 .2.1.2 Continuity equation

The PHP is tilled with a certain liquid. Alternately liquid slugs and vapour plugs are formed. When heated on one side and cooled on the other, evaporation and condensation of the fluid will take place. Evaporation takes place through a thin liquid film, which covers the inside of the tube. The continuity equation fora vapour plug is given by:

dmvi . .

- - · =m _{dt ln}. . _,V,l -m _OUI,V,l . (3.15) where min

.

^{V ' i} is the mass transfer due to evaporation (kg/s) and mOUt ^{' ' V i} the mass transfer due to condensation (kg/s) ofthe vapour plug i.

The continuity equation of a liquid slug is then

dm^{1 i (} dmv ,. dmv ^{i+l )} - -· =0.5· - -· + .

dt dt dt ^(3.16)

'

where m₁,ds the mass (kg) ofliquid slug i. The factor 0.5 accounts forthefact that the mass that condenses from a vapour plug transfers to the two adjoining liquid slugs. It is assumed that both adjoining liquid slugs receive the same amount of mass. Thus the change of mass of a liquid slug equals the average change of mass of the adjoining vapour plugs.

The rates ofmass transfer are calculated by the following equations [21]:

min,v,i

=

hc,ev1ld · Lev,i (Tev - Tv,i) I h Jg ^(3.17) mout,v,i

=

hc,co1ld ·Lco,i(Tv,i -Tco)/ hfg (3.18) where hc,ev is the evaporative heat transfer coefficient (W /m²K), Lev,i the length of the vapour in the evaporator (m), Tv,i the temperature ofvapourplug i (K), Tev the evaporator temperature (K) and hrg the latent heat of the working fluid (J/kg). For the mass transfer due to condensation hc,co is the condensation heat transfer coefficient (W /m²K), Lco,i the length ofvapour i in the condensor and Tco the condensor temperature (K).

The evaporative heat transfer is constant as long as one end of the vapour plug is inside the heated section. lf both ends are out of the heating section, the liquid film in the evaporator dries out and the mass transfer rate becomes zero.

3.2.1.3 Energy equation

The energy equation of a vapour plug equals:

dmv,iuv,i

= m .

^{.h. . -}

m

^{.h . - p . dVv,i}

where Cv is the specific heat at constant volume (J/kgK) and X is the position of the vapour plug (m) and Ris the gas constant (J/kgK).

It is assumed that vapour can be treated like an i deal gas. The pressure of the vapour plug can now be deterrnined from the ideal gas law:

(3.21)

3.2.1.4 Numerical procedure

The new values at one timestep ahead, t+t\t can be found explicitly from the old values at time t using the following equations:

mne:"_v,z =m _v,z

. +(m

_zn

.. -

_,v,z

m

_out,v,z ^\Af _P (3.22) The position X ofthe vapour plugsis determined as follows:

x;;: =

X,e,i + v1,;t\t ^(3.26)

x,ne~ e,1

=x,

e,z

.

⁺

v,

, ^('-l)1 ^t\t (3.27) where is the position of the right end of vapour plug i(m) and X1e,i the position of the left end of the vapour plug.

The time step that is used is 5* 1

o-

^{6 s.}

3.2.2 Results

Due to the oscillations of the working fluid, heat will be transferred from the hot to the cold side. Since a steady oscillation is obtained by the model the oscillation frequency will be examined.

The model can be used to determine which of the forces are dominating the behaviour of the fluid in the PHP.

Flirthermore the effect of fluid parameters can be examined. From experimental

experience Khandekar [19] suggests that a PHP working fluid should have the following properties:

• Low viscosity: lower viscosity results in lower shear stress.

• Low latent heat: when the latent heat is lower the mass flux between vapour and liquid is higher. As aresult vapour plugs will gain a higher pressure at the evaporator si de, which can result in movement of the liquid slugs.

• Low surface tension: Surface tension can create an additional pressure drop.

Oscillating motion ofthe fluid is seen to occur for PHP's with only two turns[17]. To reduce computation time not all turns of the experiment are modelled. The model geometry is depicted in tigure 3-3. The lengthof a channel from top to the bottorn of the turn is 0.1m as in the experiment. The evaporator and condensor are modelled as

isothermal surfaces with equallengths, while the diameter is equal to the hydraulic diameter in the experiment.

Temperature evaporator (⁰C) 100

Temperature condensor (°C) 20

Start temperature of vapour plugs (⁰C) 20

Totallength tube (m) 0.4

Lengthof evaporator sections (m) 0.05, 0.1 and 0.05

Lengthof condensor sections (m) 0.1

Diameter (m) 1.5* 1

o-j

Number ofplugs 3

Liquid volumeffotal volume 0.45

Heat transfer coefficient at heating wall (W/m²K) r211 150 Heat transfer coefficient at cooling wall (W /mLK) [21] 100 Table 3 Data used in model calculations

In tigure 3-7 the outcome of the calculations can be seen.

40

fluid property/water property [-]

Figure 3-7 Resulting frequency as a function of fluid parameters

The red square represents the reference situation in which the properties of water have been used in the model. Water is used as a reference since it is frequently used in PHP's.

The values of latent heat, viscosity and surface tensionhave been varied. On the x-axis the relative change of one of the fluid properties is given. A value of 2 means that the fluid property is twice as large as the fluid property of water.

It is clearly seen that the effect of surface tension is negligible. The viscosity, and especially the latent heat have a major influence on the asciilation frequency.

Since fluid movement is necessary for the transferring heat in the PHP a high frequency will be favourable. A low viscosity and low latent heat would then be advisable. It is noted that the amplitude of the asciilation remains constant within 5 % and has a mean value of 1.95 cm.

The properties of three different fluids have been entered to examine the effect on the asciilation frequency. Ethanol and methanol are used because ofthe lower latent heat and in the case of methanol also lower viscosity. The fluid properties [25] are tabulated in table 4, tagether with the resulting frequency. The geometrie data is given in table 3.

Fluid Jl hfg cr p Cv ^R Psat Frequency

(Ns/m²) (kJ/kg) (N/m) _(k_g/m³⁾ (J/kgK) (J/kg) (Pa) (Hz) Water 0.0010 2380 0.074 1000 1470 462 2329 20 Ethanol 0.0012 708 0.024 792 1242 181 5839 22.2 Methanol 0.0006 1187 0.024 792 1075 244 12968 26.7 Table 4 Properties of the fluids and resulting asciilation frequency

It is seen that the lowest frequency occurs for a water tilled PHP. Although the frequency for ethanol and methanol is higher, the frequency gain is lower than what can be expected from figure 3-7. Thus it is expected that other fluid properties have an effect on the frequency too.

The evaporator temperature is also varied to determine the minimum temperature

difference between hot and cold side at which steady oscillations occur. It is seen that for methanol steady oscillations occurred at an evaporator temperature of 45 oe. For ethanol the minimum evaporator temperature is 50 oe and for water 70°e.

The forces acting on a liquid slug are also examined with water as working fluid. The forces which act upon the liquid slug, are compared to each other, and can be seen in figure 3-8.

The major forces are caused by the pressure differences and the shear stress. The capillary and gravitational forces are small compared to these two. eompared to the pressure force the capillary and gravitational forces are less than 2 % while the friction force is about 20 % of the pressure force.

0.03

Figure 3-8 Farces acting upon liquid slug

In document Eindhoven University of Technology MASTER Single and two-phase microscale cooling Eummelen, E.H.E.C. (pagina 49-59)