# Entropy generation of radial rotation convective channels

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## Abstract

The exchange of heat between two fluids is established by radial rotating pipe or a channel. The hotter fluid flows through the pipe, while the cold fluid is ambient air. Total length of pipe is made up of multiple sections of different shape and position in relation to the common axis of rotation. In such heat exchanger the hydraulic and thermal irreversibility of the hotter and colder fluid occur. Therefore, the total entropy generated within the radial rotating pipe consists of the total entropy of hotter and colder fluid, taking into account all the hydraulic and thermal irreversibility of both fluids. Finding a mathematical model of the total generated entropy is based on coupled mathematical expressions that combine hydraulic and thermal effects of both fluids with the complex geometry of the radial rotating pipe. Mathematical model follows the each section of the pipe and establishes the function between the sections, so the total generated entropy is different from section to section of the pipe. In one section of the pipe thermal irreversibility may dominate over the hydraulic irreversibility, while in another section of the pipe the situation may be reverse. In this paper, continuous analytic functions that connect sections of pipe in geometric meaning are associated with functions that describe the thermo-hydraulic effects of hotter and colder fluid. In this way, the total generated entropy of the radial rotating pipe is a continuous analytic function of any complex geometry of the rotating pipe. The above method of establishing a relationship between the continuous function of entropy with the complex geometry of the rotating pipe enables indirect monitoring of unnecessary hydraulic and thermal losses of both fluids. Therefore, continuous analytic functions of generated entropy enable analysis of hydraulic and thermal irreversibility of individual sections of pipe, as well as the possibility of improving the thermal–hydraulic performance of the rotating pipe consisting of n sections. Analytical modeling enabled establishing of a mathematical model of the total generated entropy in a radial rotating pipe, while the generated entropy of models with radial rotating pipe were determined by experimental testing, with comparisons of the achieved results.

## Keywords

Entropy Nusselt Number Entropy Generation Rotation Number Pipe Wall## List of symbols

*v*Peripheral velocity of rotation pipe (m s

^{−1})*r*Radius distance (m)

*n*Number of revolutions (min

^{−1})*R*Radius distance from pipe section to rotation axis (m)

*T*Temperature (K)

*T*_{1}Temperature of external surface of pipe (K)

*T*_{2}Ambient temperature (K)

*p*Pressure (Pa)

- \( \dot{m} \)
Mass flow (kg s

^{−1})*A*Surface area (m

^{2})- \( \dot{S} \)
Entropy generation (W K

^{−1})- Δ
*T* Temperature change (K)

*F*_{D}Drag force (N)

- Pr
Prandtl number

*d*_{1}Inner pipe diameter (m)

*Re*Reynolds number

*Ro*Rotation number

*Gr*Grashof number

- \( \hat{L} \)
Length (m)

- \( \dot{q}_{o} \)
Heat flux rate (W m

^{−2})*D*Outside diameter of pipe (m)

*i*Specific enthalpy (J kg

^{−1})*Nu*_{D}Nusselt number per diameter

*Nu*_{l}Nusselt number per length

*R*_{sf}Gas constant of secondary fluid (J kg

^{−1}K^{−1})*S*_{t}Stanton number

*c*Specific heat (J kg

^{−1}K^{−1})- \( \dot{V} \)
Volume flow (m

^{3}s^{−1})*w*Velocity (m s

^{−1})*f*_{ω}Friction factor

## Greak symbols

- ω
Angular velocity (s

^{−1})- σ
Perimeter (m)

- ρ
Density (kg m

^{−3})- α
Convective heat transfer coefficient (W m

^{−2}K^{−1})- ν
Kinematic viscosity (m

^{2}s^{−1})- μ
Dynamics viscosity (Pa s)

- β
Angle (rad)

- γ
Angle (rad)

- ψ
Angle (rad)

- ζ
Distance (m)

## Subscripts

*p*Peripheral

*pf*Primary fluid

*sf*Secondary fluid

*rm*Rotation model

*fr*Frontal surface area

*cf*Centrifugal force

- Cor
Coriolis force

*A*Cross section A

*D*Per diameter

*L*Per length

- Δ
*T* Thermal effect

- Δ
*p* Hydraulic effect

*av*Average value

- ω,
*pf* Rotation, primary fluid

*Nu*For Nusselt number

*Nu, Re*For Nusselt and Reynolds number

*AB*Pipe section AB

*pf*,*m*Primary fluid of model

*sf*,*m*Secondary fluid of model

*gen*Generated

## Superscripts

*pf*Primary fluid

*sf*Secondary fluid

*av*Average value

*in*For input of fluid

*out*For output of fluid

- –
Dimensional quantities

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