# Heat Transfer Optimization For Smooth Circular Tubes Biology Essay

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An optimization of heat transfer for smooth circular tube has been carried out to acquire minimum outlet temperature and maximum heat flux by using Multi Objective Genetic Algorithm (MOGA II). The tube diameter with range from 7 mm to 13 mm and length of tube with range from 0.5m to 1.2 m, have been varied to study the effect the fluid flow and heat transfer of circular tubes. The numerical analysis was performed by using finite element commercial code and the optimization results show that the best design of circular tube is 7 mm for the diameter and 1.2 m for the tube length which give the minimum temperature of 8.2 °C at the outlet and maximum heat flux of 16190 W/m2.

## Introduction

With the increasing demand in power and energy over the world, a lot of research has been conducted to study the small capacity system of absorption cooling systems that can be used in residential and domestic applications. However, to redesign and optimize the overall system based on minimization of refrigerant charge, the cost of production and running still a challenging task (Misra et al. 2006).

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X. Zeng et al.(2001) have conducted an experimental work of spray evaporation of ammonia by using spray nozzles. They found that spray evaporation heat transfer coefficient increases with the increase of heat flux along the tubes and results also show that square-pitch bundle has higher spray evaporation coefficient when compared with the triangular-pitch bundle at low saturation temperature. At a high saturation temperature, triangular-pitch tube is more likely to produce higher spray evaporation coefficient.

Ghajar and Tam (1995) proposed a new flow regime map for forced flow laminar, transition and turbulent in horizontal circular tube with three different entrances (reentrant, square-edged and bell-mouth) under uniform wall heat flux. The recommended map hpears to be very general for experimental but it still can be used to predict the dath developing and fully developed flows. With knowledge of Re and the parameter Gr Pr at particular x/D location the regime map can be used to identify the convection heat transfer flow regime pure forced or mixed for any three inlets.

H.A. Mohammed and Y.K. Salman (2007) investigated an experimentally laminar combined convection heat transfer to thermally developing air flow inside a uniformly horizontal circular cylinder. The effect of Reynolds number and the effect of heat flux were determined. They found that the variation of the surface temperature along the cylinder has the same shape, and it would be higher for low Re than for high Re number due to fee convection domination. It was concluded that the free convection effects tended to decrease the heat transfer results at low Re and to increase the heat transfer results for high Re.

Another study on the flow inside the tube was presented by (Lap-Mou Tam and Afshin J. Ghajar 1998). They investigated the behavior of local heat transfer coefficient for the transition region of a circular tube with a bell-mouth entrance and uniform wall heat flux around the boundary. An experimental test was carried out on the tube with inner diameter of 1.584 cm and ethylene glycol water mixture used as a working fluid. It was established that for a bell-mouth entrance the variation of local heat transfer coefficient with the length is unusual and causing a dip in the Nu-x/D curve. The length of this dip is very short (about 25 diameters) in the turbulent region, however, the length of the dip in the transition region is much longer than in the turbulent region. This phenomenon causes a significant influence on both the local and the average heat transfer characteristics of the tube.

(B. Shome and M. k. Jensen 1995) analysis the thermal developing and simultaneously developing mixed convection flow with variable viscosity for both heating and cooling in isothermal horizontal circular ducts. From their analysis, the analysis that used in this study is for uniform wall temperature boundary condition. It was found that effect of variable viscosity was more pronounced on the fraction factor than Nusselt Number. They proposed correlations that can be used for wide applications and get more accurate results.

(Alper Yilmaz 2008) Studied the optimization of heat transfer in the tube length for a turbulent flow in a smooth wall tube, at constant wall temperature for a given pressure loss. He found that for a certain tube length to diameter ratio (L/D) in turbulent flow, the maximum heat transfer flux was depended on pressure Reynolds number , Prandtl number and local pressure loss coefficient, optimum value of L/D increase with Pr and numbers. The maximum heat transfer flux increase with and decrease with Pr number.

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From the previous studies, most of the researchers have investigated the behaviours of the fluid flow inside the pipes with different inlet geometry and flow regimes for free and force convection. Several studies were also carried out to study on the effects of boundary sub layer on the heat transfer coefficient and different types of tubes bundle.

However, the outlet temperature of the tube in shell and tubes heat exchanger has yet to be investigated.

Therefore, the objective of this study is to optimize the design the circular tubes inside the evaporator by using Multi Objective Genetic Algorithm (MOGA II) method. The Navier-Stokes equation was employed to analysis the fluid flow and heat transfer inside the circular tube.

The use of shell and tube and spray evaporation with triangular tube bundle as the heat exchanger in the evaporators are wide, and the most important point is how to get the optimum design for the heat transfer and to get the lower temperatures value in the tubes inside. The effect of the amount of heat flux that gains from the chilled water inside the pipes should be studied.

In this study investigation of the fluid flow and the heat transfer inside a pipe which used in the evaporator of absorption cooling system of 1.5 Ton refrigerant has been proposed. A fixed amount of flow rate of water inside the circular tubes is studied. The study on the design parameters of different diameters and Lengths on the fluid flow and the heat transfer amount, at constant wall temperature has been studied.

## The study on the design parameter

The study on the design parameters for the tubes inside the shell and tube heat exchanger for an evaporator used in the absorption cooling system is explain below. The effect of tube diameter and the length of the tube is been studied to get the optimum heat flux from the tube boundary and to get the minimum temperature form the tube outside. The effect of design parameter is studied for constant mass flow rate inside the tube, with variable physical properties depends on the temperature. The water is used inside the pipes; the flow inside the tube is laminar with different Reynolds.

μ, u,T, In Water Out μ, u,T,

## Mathematical model

This study focuses on the optimization of the fluid flow and the heat transfer inside circular tube. The equations that used in this study to analysis the heat and fluid flow are:

## The Energy equation

The fundamental law governing for the heat transfer is the first law of thermodynamics, usually referred to as the principle of conservation of energy. Therefore the basic law is usually rewritten in term of temperature, T. For a fluid, the resulting heat equation is

(1-1)

Where

Is the density (Kg/m3)

Is the specific heat capacity at constant pressure (j/ (Kg.K))

Is absolute temperature (k)

Is the velocity vector (m/s)

Is the heat flux by conduction (W/m2)

Is the pressure (pa)

Is the viscous stress tensor (pa)

Is the strain rate tensor (1/s):

Contains heat sources other than viscous heating (W/m3 )

By using Fourier's Law of conductions, which states that conductive heat flux, q, is proportional to the temperature gradient:

(1-2)

Inserting equation 1-2 into equation 1-1, reordering the terms and ignoring viscous heating and pressure work puts the heat equation on this form:

(1-3)

## The Continuity and Momentum Equations :

The equations below represent continuity equation that refer to the conservation of mass, and the second equation is the conservation of momentum and the third one represented the conservation of energy, these three equations represent The Navier-Stokes equation used for the single-phase fluid- flow

(1-4)

(1-5)

(1-6)

Where

Is the density (Kg/m3)

Is the velocity vector (m/s)

Is pressure (pa)

Is the viscous stress tensor (pa)

Is the volume force vector (N/m3)

Is the specific heat capacity at constant pressure (j/(kg.K))

Is the absolute temperature (k)

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Examples of our workIs the heat flux vector (W/m2)

Contains the heat sources (W/m3)

Is the strain rate tensor: =)

## Boundary conditions

The boundary condition assumption for the present study is presented, for the tube inlet the temperature of the fluid (water) is 293.15oK and the average mean velocity from 0.34 m/s to 0.0662 m/s according to the different of inner diameter from 7mm to 13mm. the temperature of the outlet wall of the tube is constant and it's equal to 278.15oK. The convective flux boundary condition applied at the outflow from the circular tube, this condition states that the only heat transfer over a boundary is by convection, the temperature gradient in the normal direction is zero and there is no radiation (COMSOL user's guide heat transfer module). No-slip condition of the flow on the inner wall pipe was applied. The fluid is Newtonian, Incompressible, with transport properties in depended of time and position. The physical properties for the fluid inside the pipe depend on Temperature using the Liquids and gases material library by COMSOL MULTIPHYSICS. The flow inside the pipe is laminar and developed with different Reynolds Number from 2089 to 1141. Incompressible Navier-Stockes equation with Convection and conduction module is used in the analytic simulation. The tube with Different Diameter between (7-16 m) mm and the length of tubes from (0.5-1.2 m) is used and the overall flow rate is 0.1197 kg/s. the constant physical properties of material used for the pipes. The material used for the pipe is [Steel AISA 4340].

## Numerical method

A COMSOL Multiphysics 3.5a is employed to solve this case of study. It is a powerful interactive environment for modeling and solving all engineering and scientific problem based on partial differential equations. COMSOL Multiphysics used Finite Element Method to solve the model. The software runs the finite element analysis together with adaptive meshing and error by using different numerical solver (user's guide comsol multiphysics). . The solver that used is stationary analysis type and the linear system solver type used direct (SPOOLES).

By using extra fine meshing the number of degree of freedom are 536118 and number of mesh points 25443. Fig. 2. Show the mesh statistics, and fig. 3. The temperature profile and the velocity profile for the fluid flow inside a 3D axisymmetric pipe model.

Fig. 2. Shows the mesh statistics

Fig. 3. Shows the profile temperature and velocity for the flow inside the pipe

## Results and discussion for the mathematical model simulation

The effect of the Diameter and Length of the tube on the heat transfer and fluid flow inside the pipe has studied. As mention previously the flow inside the pipe is Laminar and the case study assumes the flow inside the tube is not fully development according to the calculation of hydrodynamic entry length calculations. The effect of Diameter and Length of the pipe are discussed.

## Effect of Tube diameter and Length on the heat flux

The heat Flux results present in Figs. 4 and Fig. 5 for different Lengths and Diameters , the results show that the best heat flux within the smallest diameter 7 mm and shortest length 0.5 m of the pipe, because of high Reynolds number with high velocity. And the reason of the shortest length pipe is referred to the constant flow rate through the pipe. So the temperature difference between the inlet and the out let will be higher, according to the Newton's law of cooling for constant wall temperature and constant heat transfer coefficient. Therefore as the pipe length increases the heat flux will decrease. Also from Figs. 4 the heat flux increase again especially for the large tube diameter 10-13 mm this phenomena because the flow will be hydrodynamically fully development and the difference between the bulk temperature and surface temperature increase and simultaneously the heat flux increase.

Fig. 4. Variation of heat flux at different Tube Diameter and Length

Fig. 5. Variation of heat flux at Different Tube Diameter and Length

## Effect of Tube diameter and Length on the outlet temperature

In fig. 6. And fig. 7. the temperature outlet of the fluid at different Diameter and tube length is shown , the minimum temperature got within the smallest tube diameter 7 (mm) and Longest Length of the tube 1.2 (m) the reason behind that is for the highest Reynolds number and largest surface area that conducted better heat transfer. Also from the Figs it is obvious that the effects of the varying Length in small diameters are beggar than the large diameters.

Fig. 6. Variation of out let temperature with Different Tube Diameter and Length

Fig. 7. Variation of out let temperature with Different Tube Diameter and Length

## Effect of Tube diameter and Length on the Pressure Drop

The pressure drops along the tube for different diameter has been presented. In Figs. 8. The pressure drop is a quantity of interest in the analysis of pipe flow. It is directly related to the power consume in the pump. In this study the effect of the Diameter and the length are explain for the constant mass flow rate inside the tube, from the figure the max pressure drop appear in the smallest diameter and longest tube and it is 7 (mm) Diameter and 1.2 (m) this phenomenon can explain according to the Darcy equation

(2-1)

Where is the dynamic pressure and is the Darcy friction factor.

Fig. 8. Variation of Pressure drop with Different Tube Diameter and Length

## Optimization

Nowadays there is a trend to increase the robustness and performance of the design. The aim of the optimization is to increase the performance of the parameters that used in the design. In the present work the optimization task has been carried out, by using the powerful optimization software namely Mode FRONTIER. There are two design parameters taken into account and three main objectives that needed to be achieved. Multi Objective Genetic Algorithm (MOGA-II) has been used to achieve a set of optimal solution. MOGA-II proposed a set of alternative optimum solution named the Pareto frontier (Augusto et al. 2006). The objective of the optimization is to maximize the heat flux along the tube and maximize the temperature different between the inlet and outlet flow, with logical pressure drop.

To start the optimization process, there is a parameters should be arranged and setting in modeFRONTIER for the desired design. Starting with initial design of experiment (DOE) which consist of top slant angle and bottom slant angle had been generated by using Face Centered Cubic (FCC) method. The selection of the FCC method was to generate a good starting point for multi-objective genetic algorithm (MOGA-II) which is available in mode FRONTIER. By using the Data Wizard Tool, the initial results of simulations were imported into the Mode FRONTIER. The optimization scheme was automatically generated as shown in Fig.9.

Fig. 9 ModeFRONTIER optimization scheme

Before the multi-objective genetic algorithm (MOGA-II) run. The imported results were interpolated by Response Surface Modeling (RSM) based on the Gaussian Processes algorithm. Where (RSM) is a collection of mathematical and statistical techniques useful for the modeling and analysis of problems in which a response of interest is influenced by several variable. The Gaussian Processes algorithm is a powerful regression model. This method is best suited for non-polynomial responses. After that the multi-objective genetic algorithm (MOGA-II) has been run. In the final step of the optimization process the Multi Criteria Decision Making (MCDM) will be used. The MCDM assist the decision maker in finding the best solution from among a set of reasonable alternatives. In the MCDM attributes panel the decision quantity has been done (the input or output of the project and there relations). The aim of the objectives is to maximize the heat flux, minimize the temperature out and the pressure drop along the tube. The weight of these objectives is different according to the significance of these factors. In the present work the effect of the pressure drop is negligible comparing with the effect of heat flux and temperature drop. Because of the small amount of flow rate, so the effect of the head pressure with power consuming is small when you compare with the effect of increase in heat flux. The heat flux is important in this study to insure that the saturated liquid ammonia can be evaporated and changed to the saturated vapor by get the amount of heat. The weight of the temperature is more than the heat flux variable because in this model we need the minimum temperature out of the chilled water that supply to the fan coils used in the air conditioning process and this is the aim of good evaporator. From above the assumption in the MCDM attribute panel is:

Temperature out Ëƒ 2.00 heat flux,

Heat Flux Ëƒ 2.00 Pressure drop,

Temperature out Ëƒ 4.00 Pressure drop.

Fig. 10. Shows the MCDM utility chart, also shows the weights of different objectives.

Fig. 10. MCDM utility chart

There are 1918 design points were generated by MOGA-II analysis from initial 19 initial populations. To make the final solution MCDM was used. The results of MCDM show that the best design that give the higher rank is the design 1 (the tube diameter 7mm and the Length 1.2 m). Fig 11 shows the Rank of the given designs.

Fig 11. The Rank of the given design

## Conclusion

Numerical analysis and optimization of heat transfer and fluid flow for circular smooth tube has been studied. The effect of the tube diameter and length was studied. It was found that the maximum heat flux occurs within smallest tube diameter and shortest length for constant wall temperature at constant flow rate. The temperature out from the tube has been presented and the lowest value founded within the smallest tube diameter 7(mm) and longest length of the tube 1.2(m). The maximum pressure drop appears in the smallest diameter and longest tube. It was concluded that the best design for the pipes has been found with 7(mm) diameter and 1.2(m) for 0.0133 (kg/s) by using Multi Objective Genetic Algorithm. This method gave a set of optimal solution with different rank according to the weight of the input variables and objectives.