# Soft computing and statistical approach for the sensitivity analysis of heat transfer through the hybrid nanoliquid film in a 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 around its own axis at a speed of (Omega .) Speed (left(overrightarrow{V}right)) the components in the 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.

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

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, the centrifugal force ({Omega }^{2}R) acts against the gravitational force in the Y direction (Fig. 1). So Eq. (1) and (2) as velocity components are given by:

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

(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 area via the adiabatic zone. In the evaporator area, the liquid film takes the heat from the evaporator and the liquid evaporates. These vapors return to the condenser area and condense back into the liquid. In this analysis speed component ({u}_{y}) is considered negligible. The inertia of the hybrid nanoliquid flow is assumed to be very small compared to other forces. The linear mass flow rate is assumed to be zero at both ends of the pipe and no slip condition is considered at the pipe wall. The thickness of the hybrid nanoliquid film (zeta (x)) is very small compared to the radius of the pipe.

Velocity and mass flow boundary conditions

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

(6)

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

(seven)

At the (liquid/vapor) boundary of the 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)

here ({widehat{M}}_{hnl}) is the mass flow rate of the hybrid nanoliquid (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 speed, the pressure term can be eliminated from the equations. (4) and (5) and therefore the velocity of the hybrid nanoliquid (({u}_{x})) can be expressed as follows:

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

(ten)

For high-speed rotations, the terms ({tau}_{v}) and ({P}_{zeta }) are negligible compared to the other terms (Song et al.4 also steam velocity (({widehat{omega }}_{vap})) at the liquid/vapor boundary is much greater than the velocity of the nanoliquid ({u}_{x, zeta})So ({widehat{omega }}_{vap}+{u}_{x, zeta }approx {widehat{omega }}_{vap})

By using eq. (9) the unit width of the film, the hybrid nanoliquid flow rate is given by:

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

(11)

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

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

(12)

Temperature boundary conditions

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

(13)

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

(14)

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

(15)

(tau) is the thickness of the pipe.

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

(16)

Here, the condenser wall temperature is assumed to be constant along the length. It is also assumed that the heat flux 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 boundary conditions (13)-(16), the total heat flux 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 on the mean phase change enthalpy and given as

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

(18)

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

Compare eq. (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)

The heat input to the pipe through the evaporator section is used to convert the liquid into a vapor state, hence

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

(20)

By 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)

Combine eq. (19) and (20) we obtain:

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

(22)

By using eq. (19) in (11) the change in thickness of the hybrid nanoliquid film can be expressed as:

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

(23)

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

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

Next Waini et al.40the thermal and physical properties of the 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 the concentration of nanoparticles in a pure liquid, (rho) is the density and (C_{p}) is the specific heat. The suffixes “p1”, “p2” and “f” represent respectively the GO nanoparticle, the MoS2 nanoparticle and the pure fluid.

As nanoparticles are not considered in the vapor phase, so the enthalpy of phase change (Delta H) of the hybrid nano-fluid will be due to the pure fluid only and given by the following relationship:((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 the nanoparticles are shown in Table 1. 