# Soft computing and statistical approach for sensitivity analysis of heat transfer through the hybrid nanoliquid film in rotating heat pipe

In the present mathematical model, the two dimensional laminar flow of hybrid nanoliquid is considered inside the thin film region, which is generated due to the high speed rotation of the cylindrical pipe. The x-axis is taken as the axis of the pipe which rotates about its own axis at a rate of (Omega .) Velocity (left(overrightarrow{V}right)) components in X and Y directions are ({u}_{x}) and ({u}_{y}) respectively. The physical mode of the pipe and the coordinate system are shown in Fig. 1.

Following Uddin et al.16 the equations for the flow of hybrid nanoliquid inside heat pipe are given as:

Continuity equation:

$$overrightarrow{nabla }.overrightarrow{V}=0$$

(1)

Momentum equation:

$$left(overrightarrow{V}.overrightarrow{nabla }right)overrightarrow{V}=-frac{1}{{rho }_{hnl}}overrightarrow{nabla }P+ frac{{mu }_{hnl}}{{rho }_{hnl}}{nabla }^{2}overrightarrow{V}+overrightarrow{F}$$

(2)

Due to the rotation of the pipe, centrifugal force ({Omega }^{2}R) acts against the gravitational force in Y-direction (Fig. 1). Therefore Eq. (1) and (2) in velocity components form are given as:

$$frac{partial {u}_{x}}{partial x}+frac{partial {u}_{y}}{partial y}=0$$

(3)

$${u}_{x}frac{partial {u}_{x}}{partial x}+{v}_{y}frac{partial {u}_{x}}{ partial y}=-frac{1}{{rho }_{hnl}}frac{partial P}{partial x}+frac{{mu }_{hnl}}{{rho } _{hnl}}left(frac{{partial }^{2}{u}_{x}}{partial {x}^{2}}+frac{{partial }^{2} {u}_{x}}{partial {y}^{2}}right)$$

(4)

$${u}_{x}frac{partial {u}_{y}}{partial x}+{v}_{y}frac{partial {u}_{y}}{ partial y}=-frac{1}{{rho }_{hnl}}frac{partial P}{partial y}+frac{{mu }_{hnl}}{{rho } _{hnl}}left(frac{{partial }^{2}{u}_{y}}{partial {x}^{2}}+frac{{partial }^{2} {u}_{y}}{partial {y}^{2}}right)+left(g-{Omega }^{2}Rright)$$

(5)

Practically, in rotating heat pipes the liquid flows from the condenser to the evaporator zone via the adiabatic zone. In the evaporator zone, the liquid film takes the heat from the evaporator and the liquid gets evaporated. These vapors travel back to the condenser zone and get condensed into the liquid again. In the present analysis velocity component ({u}_{y}) is considered negligible. The inertia in the flow of hybrid nanoliquid is assumed to be very small compared to other forces. The rate of linear mass flow is assumed to be zero at both ends of the pipe and no slip condition is considered at the wall of the pipe. The hybrid nanoliquid film thickness (zeta (x)) is very small compared to the radius of the pipe.

Velocity and mass flow boundary conditions

$${text{At extremities of the pipe}}:{text{ at}};x = 0;and;x = {mathcal{L}};{text{Fluid mass flow rate }},;hat{M}_{hnl} = 0$$

(6)

$${text{At }};{text{the }};{text{wall }};{text{of}};{text{ the}};{text { pipe}}:{text{ at}};y = 0,;u_{x} = 0;left( {{text{No }};{text{slip }}; {text{condition}}} right)$$

(7)

At the boundary (liquid/vapor) of hybrid nanoliquid film (Daniels et al.2):

$$At;y = zeta left( x right), P_{liquid} = P_{vapor} = P_{sat} and mu_{hnl} frac{{partial u_{x} }}{ partial y} = – tau_{v} – left( {hat{omega }_{vap} + u_{x, zeta } } right)frac{{dhat{M}_{ hnl} }}{dx}$$

(8)

TIMES ({widehat{M}}_{hnl}) is the hybrid nanoliquid mass flow rate (Linear/per unit width) and given by

$${widehat{M}}_{hnl}={int }_{0}^{zeta }{rho }_{hnl}{u}_{x}dy$$

(9)

Using the above assumptions and boundary conditions (6)-(8), and following Uddin et al.16 the velocity the pressure term can be eliminated from Eqs. (4) and (5) and hence the velocity of the hybrid nanoliquid (({u}_{x})) can be expressed as:

$${u}_{x}=-frac{1}{{mu }_{hnl}}frac{partial {P}_{zeta }}{partial x}left(frac {{y}^{2}}{2}-zeta .yright)-frac{{rho }_{hnl}}{{mu }_{hnl}}left({Omega } ^{2}Rgright)frac{partial zeta }{partial x}left(frac{{y}^{2}}{2}-zeta .yright)-frac{ y}{{mu }_{hnl}}{tau }_{v}-frac{y}{{mu }_{hnl}}frac{d{widehat{M}}_{hnl }}{dx}left({widehat{omega }}_{vap}+{u}_{x, zeta }right)$$

(10)

For high speed rotations the terms ({tau }_{v}) and ({P}_{zeta }) are negligibly small with respect to other terms (Song et al.4 also the vapor velocity (({widehat{omega }}_{vap})) at the liquid/vapor boundary is much larger than the nanoliquid velocity ({u}_{x, zeta })therefore ({widehat{omega }}_{vap}+{u}_{x, zeta }approx {widehat{omega }}_{vap})

Using Eq. (9) the per unit width of the film, the hybrid nanoliquid flow rate is given as:

$${widehat{M}}_{hnl}=frac{{rho }_{hnl}^{2}}{{mu }_{hnl}}left(g-{Omega }^ {2}Rright)frac{{zeta }^{3}}{3}left(frac{partial zeta }{partial x}right)-frac{{rho }_ {hnl}}{{mu }_{hnl}}frac{d{widehat{M}}_{hnl}}{dx}{widehat{omega }}_{vap}frac{{ zeta }^{2}}{2}$$

(11)

In hybrid nanoliquid film, the heat equation is written as:

$${u}_{x}frac{partial theta }{partial x}+{v}_{y}frac{partial theta }{partial y}=frac{{k} _{hnl}}{{rho }_{hnl}{Cp}_{hnl}}left(frac{{partial }^{2}theta }{partial {x}^{2}} +frac{{partial }^{2}theta }{partial {y}^{2}}right)$$

(12)

Temperature boundary conditions

$${text{At }};{text{extremities }};{text{of}};{text{ the}};{text{ pipe}}: , ; {text{at}};x = 0;{text{and}};x = {mathcal{L}},;theta = 0$$

(13)

$${text{At}};{text{ the}};{text{ inner}};{text{ wall }};{text{of }};{text {the}};{text{ pipe}}:{text{at}};y = 0,;theta = theta_{w} ;and;k_{hnl} left( { frac{partial theta }{{partial y}}} right) = H_{1} left( x right)$$

(14)

$${text{At }};{text{the }};{text{outer }};{text{wall }};{text{of }};{text {the}};{text{ pipe}}: {text{at}};y = tau ,;theta = theta_{w} ;and;k_{Cu} left( {frac{partial theta }{{partial y}}} right) = H_{2} left( x right)$$

(15)

(tau) is the thickness of the pipe.

$${text{At}};{text{ the }};{text{boundary}};{text{ of}};{text{ hybrid }};{text {nanoliquid}};{text{ film}}:{text{ At}};y = zeta left( x right),;theta = theta_{s}$$

(16)

Here the condenser wall temperature is assumed to be constant over the length. It is also assumed that, the heat flow is only due to the condensation/evaporation of the hybrid nanoliquid film and the pipe wall in the direction perpendicular to the pipe axis.

Therefore, using Eq. (12) and the boundary conditions (13)-(16), the total heat flow per unit circumference of the pipe can be expressed as:

$$Hleft(xright)=left({theta }_{w}-{theta }_{s}right)/left(frac{zeta left(xright) }{{k}_{hnl}}+frac{tau }{{k}_{Cu}}right)$$

(17)

Following Daniels et al.2, (Hleft(xright)) depends upon the average phase change enthalpy and given as

$$Hleft(xright)=-widehat{mathrm{Delta h}}frac{d{widehat{M}}_{hnl}}{dx}$$

(18)

TIMES (widehat{mathrm{Delta h}}=mathrm{Delta h}+0.35Cp.left(Delta theta right)) is the average enthalpy of vaporization.

Comparing Eqs. (17 and 18), gives

$$frac{d{widehat{M}}_{hnl}}{dx}=-left({theta }_{w}-{theta }_{s}right)/widehat{ mathrm{Delta h}}left(frac{zeta left(xright)}{{k}_{hnl}}+frac{tau }{{k}_{Cu}} right)$$

(19)

Heat input to the pipe through the evaporator section is utilized in converting the liquid into vapor state, therefore

$$Hleft(eright)={rho }_{v}widehat{mathrm{Delta h}}{widehat{omega }}_{vap}$$

(20)

Using Eq. (17)

$$Hleft(eright)=Hleft(x:0le xle {mathcal{L}}_{e}right)=left({theta }_{e}- {theta }_{s}right)/left(frac{zeta left(xright)}{{k}_{hnl}}+frac{tau }{{k}_{ Cu}}right)$$

(21)

Combining Eqs. (19) and (20) we get:

$${widehat{omega }}_{vap}=left({theta }_{e}-{theta }_{s}right)/{rho }_{v}widehat{ mathrm{Delta h}}left(frac{zeta left(xright)}{{k}_{hnl}}+frac{tau }{{k}_{Cu}} right)$$

(22)

Using Eq. (19) in (11) the hybrid nanoliquid film thickness change can be expressed as:

$$frac{partial zeta }{partial x}=frac{3{mu }_{hnl}{widehat{M}}_{hnl}}{{rho }_{hnl}^ {2}left(g-{Omega }^{2}Rright){zeta }^{3}}-frac{3{widehat{omega }}_{vap}left({ theta }_{w}-{theta }_{s}right)}{2{rho }_{hnl}left(g-{Omega }^{2}Rright)zeta widehat{Delta h}left(frac{zeta }{{k}_{hnl}}+frac{tau }{{k}_{Cu}}right)}$$

(23)

The net heat flux can be calculated by using the formula: (HeatFlux = 2pi r(Delta H)_{v} .overset{lower0.5emhbox{$smash{scriptscriptstylefrown}$}}{M}_{hnl}).

TIMES ((Delta H)_{v}) is enthalpy of vaporization.

Following Waini et al.40the thermal and physical properties of hybrid nanoliquid are given by:

$${text{Density}}:;rho_{hnl} = (1 – phi_{2} )[phi_{1} rho_{p1} + (1 – phi_{1} )rho_{f} ] + phi_{2} rho_{p2}$$

(24)

$${text{Specific heat capacity}}:;(rho C_{p} )_{hnl} = (1 – phi_{2} )[phi_{1} (rho C_{p} )_{p1} + (1 – phi_{1} )(rho C_{p} )_{f} ] + phi_{2} (rho C_{p} )_{p2}$$

(25)

$${text{Dynamic Viscosity}}: ;mu_{hnl} = frac{{mu_{f} }}{{[(1 – phi_{1} )(1 – phi_{2} )]^{2.5} }}$$

(26)

Thermal Conductivity

$$k_{nl} = frac{{k_{p1} + 2k_{f} – 2phi_{1} (k_{f} – k_{p1} )}}{{k_{p1} + 2k_{f } + phi_{1} (k_{f} – k_{p1} )}}k_{f} ;and;k_{hnl} = frac{{k_{p2} + 2k_{nl} – 2 phi_{2} (k_{nl} – k_{p2} )}}{{k_{p2} + 2k_{nl} + phi_{2} (k_{nl} – k_{p2} )}}k_{nl }$$

(27)

where (phi) is nano-particle concentration in pure liquid, (rho) is the density and (C_{p}) is the specific heat. The suffixes “p1”, “p2” and “f” are representing the GO-nanoparticle, MoS2-nanoparticle and pure fluid respectively.

Since the nano-particles are not considered in vapor phase, therefore the phase change enthalpy (Delta H) of the hybridnano-fluid will be due to the pure fluid only and given by the following relation:((rho Delta H)_{hnl} = (1 – phi_{hnl} )(rho Delta H)_{f})where (phi_{hnl} = phi_{1} + phi_{2}). The properties of the working fluid and nanoparticles are illustrated in Table1.