Two-Dimensional Analytical Solution of the Laminar Forced Convection in a Circular Duct with Periodic Boundary Condition

Abstract

In the present study analytical solution for forced convection heat transfer in a circular duct with a special boundary condition has been presented, because the external wall temperature is a periodic function of axial direction. Local energy balance equation is written with reference to the fully developed regime. Also governing equations are two-dimensionally solved, and the effect of duct wall thickness has been considered. The temperature distribution of fluid and solid phases is assumed as a periodic function of axial direction and finally temperature distribution in the flow field, solid wall, and local Nusselt number, is obtained analytically.

1. Introduction

In some convection and conduction heat transfer application, the temperature of boundaries is not constant and is a function of a dimension (X, Y, Z) or time. Some of these functions can be modeled as a periodic function. The mentioned boundary condition is applied in several engineering problems such as the nuclear reactor cooling system design and Stirling-cycle machines heat exchanger. Also in many industrial boilers which the heating up process is done by several burners, the above boundary condition is useful for numerical or analytical simulation.

Barletta et al. [1, 2] studied forced convection flow, respectively, with sinusoidal wall heat flux distribution and sinusoidal temperature distribution on the longitudinal boundary in a plane channel. Mentioned studies revealed that temperature distribution in longitudinal direction can be expressed as aperiodic function of the longitudinal coordinate. Also in [3] the effect of viscous dissipation on temperature in a circular duct in the case of a sinusoidal axial variation of the wall heat flux is evaluated.

In [4, 5] the laminar forced convection with the axial periodic boundary condition is studied by taking into account the axial heat conduction. Barletta et al. [6] investigated the effect of conducting wall in a parallel-plane channel. Their study showed that the increase in wall heat conductivity causes the increase in temperature oscillation amplitude off low field.

In addition to the analytical studies, some numerical studies are done on convection heat transfer with periodic boundary condition [7–11]. Periodic forced convection in the thermal entrance region of a porous medium is investigated in [12] by using finite element method. In mentioned that study, a two-temperature model employed in order to evaluate the solid and fluid phase temperatures.

In the present paper, the forced convection in a circular duct with the boundary condition given by a temperature distribution which varies axially with sinusoidal law is solved analytically. It should be mentioned the modeling is done by taking into account the heat conduction in the duct wall. The longitudinal heat conduction and the viscous dissipation in the fluid will be neglected. Reference will be made to the hydrodynamically and thermally developed region. The two-dimensional energy balance equations will be solved analytically both for the fluid and the solid region.

The fluid and solid temperature distribution will be obtained analytically by expressing the energy balance equation as a complex-valued hypergeometric confluent equation.

For the analytical solution, [2] is used as the main source.

2. Mathematic Model

In this section, the local energy balance equations, written separately for the solid region and for the fluid region, are solved analytically with the mentioned boundary condition.

Let us consider, an infinite circular duct with internal and external radii is, respectively, 𝑟₀, 𝑟₁. Since the channel has a symmetry with reference to the 𝑟=0, it is possible to study only half of the solution domain (0≤𝑟≤𝑟₁). A drawing of the section of the circular duct together with the boundary condition is shown in Figure 1.

Figure 1 

The axial section of channel.

With considering the fully developed convection of Newtonian fluid, constant thermophysical property, neglecting of axial heat conduction in both phase and viscose dissipation in the fluid phase, the local energy balance in fluid and solid phases is given by the following.

Fluid phase (0<𝑟<𝑟₀): 𝜕𝑟𝜕𝑟𝜕𝑇−𝑈(𝑟)𝜕𝑟𝛼_𝑓𝑟𝜕𝑇𝜕𝑥=0. (1) Solid phase (𝑟₀<𝑟<𝑟₁): 1𝑟𝜕𝑟𝜕𝑟𝜕𝑇+𝜕𝜕𝑟²𝑇𝜕𝑥²=0, (2) where 𝑈(𝑟) is the Poiseuille velocity distribution: 𝑈(𝑟)=2𝑢𝑟1−𝑟₀². (3) The equations must be solved together with following boundary conditions: at𝑟=0⇒𝜕𝑇𝜕𝑟=0,at𝑟=𝑟₁⇒𝑇=𝑇₀+Δ𝑇sin𝛽𝑥,at𝑟=𝑟₀⇒𝑇_𝑠=𝑇_𝑓,at𝑟=𝑟₀⇒𝑘_𝑠𝜕𝑇_𝑠𝜕𝑟=𝑘_𝑓𝜕𝑇_𝑓.𝜕𝑟 (4) With introducing the following dimensionless parameters: 𝑟𝜂=𝑟₀𝑥,𝜉=𝑟₀,𝜃=𝑇−𝑇₀𝑟Δ𝑇,𝜎=¹𝑟₀𝑘,𝛾=^𝑠𝑘_𝑓. (5) Equations (1), (2), and (4), respectively, transform to 𝜕𝜂𝜕𝜂𝜕𝜃𝜕𝜂−Pe1−𝜂²𝜂𝜕𝜃1𝜕𝜉=0,0<𝜂<1,𝜂𝜕𝜂𝜕𝜂𝜕𝜃+𝜕𝜕𝜂²𝜃𝜕𝜉²=0,1<𝜂<𝜎,at𝜂=0⇒𝜕𝜃_𝑓𝜕𝜂=0,at𝜂=𝜎⇒𝜃=sin𝐵𝜉,at𝜂=1⇒𝜃_𝑠=𝜃_𝑓,at𝜂=1⇒𝑘_𝑠𝜕𝜃_𝑠𝜕𝜂=𝑘_𝑓𝜕𝜃_𝑓⇒𝜕𝜂𝜕𝜃_𝑓𝜕𝜂=𝛾𝜕𝜃_𝑠,𝜕𝜂 (6)(7) where Pe=2𝑟₀(𝑢/𝛼_𝑓),𝐵=𝛽𝑟₀.

3. Analytical Solution

By assuming the axial position sufficiently far from the inlet section, the solution of (6) can be written as 𝜃𝜉,𝜂=𝜃₀𝜂+𝜃₁𝜂sin(𝐵𝜉)+𝜃₂𝜂cos(𝐵𝜉). (8) By replacing (8) into (6), the 𝜃₀(𝜂),𝜃₁(𝜂),and𝜃₂(𝜂) can be obtained from the following equations sets: 𝑑0<𝜂<1⇒𝜂𝑑𝜂𝑑𝜃₁𝑑𝜂+Pe𝐵1−𝜂²𝜂𝜃₂𝑑=0𝜂𝑑𝜂𝑑𝜃₂𝑑𝜂−Pe𝐵1−𝜂²𝜂𝜃₁𝑑=0𝜂𝑑𝜂𝑑𝜃₀1𝑑𝜂=0,1<𝜂<𝜎⇒𝜂𝑑𝜂𝑑𝜂𝑑𝜃₁𝑑𝜂−𝐵²𝜃₁1=0𝜂𝑑𝜂𝑑𝜂𝑑𝜃₂𝑑𝜂−𝐵²𝜃₂1=0𝜂𝑑𝜂𝑑𝜂𝑑𝜃₀𝑑𝜂=0. (9) And the boundary condition of equations sets (9) is given by 𝜂=0→𝑑𝜃₀=𝑑𝜂𝑑𝜃₁=𝑑𝜂𝑑𝜃₂𝑑𝜂=0,𝜂=1⇒𝑑𝜃_0,𝑓𝑑𝜂=𝛾𝑑𝜃_0,𝑠𝑑𝜂𝑑𝜃_1,𝑓𝑑𝜂=𝛾𝑑𝜃_1,𝑠𝑑𝜂𝑑𝜃_2,𝑓𝑑𝜂=𝛾𝑑𝜃_2,𝑠𝜃𝑑𝜂_0,𝑓=𝜃_0,𝑠𝜃_1,𝑓=𝜃_1,𝑠𝜃_2,𝑓=𝜃_2,𝑠,𝜂=𝜎⇒𝜃₀=0,𝜃₁=1,𝜃₂=0. (10) In (9) it can be observed 𝜃₁,𝜃₂ are coupled together, and 𝜃₀ is independent of 𝜃₁,𝜃₂. The solution of boundary valued problem (9) with considering boundary conditions (10) yields 𝜃₀=0.

For uncoupling the 𝜃₁,𝜃₂ equations, a new function 𝜓 is introduced as 𝜓=𝜃₁+𝑖𝜃₂. (11) By employing (11), (9) can be collapsed into a boundary valued problem, such as (12) 10<𝜂<1⇒𝜂𝑑𝜂𝑑𝜂𝑑𝜓𝑑𝜂−𝑖Pe𝐵1−𝜂²1𝜓=0,1<𝜂<𝜎⇒𝜂𝑑𝜂𝑑𝜂𝑑𝜓𝑑𝜂−𝐵²𝜓=0. (12)(13) Also boundary conditions (10) transform to 𝜂=𝜎⇒𝜓(𝜎)=1,𝜂=0⇒𝑑𝜓𝑑𝜂=0,𝜂=1⇒𝜓_𝑓=𝜓_𝑠,𝜂=1⇒𝑑𝜓_𝑓𝑑𝜂=𝛾𝑑𝜓_𝑠.𝑑𝜂 (14) By introducing the new parameters 𝜓=Φ/𝜂, 𝜔=√−𝑖Pe𝐵/2, and 𝑧=2𝜔𝜂² and replacing into (12), equation (15) is formed as Φ^′′+14𝑧²+𝜔−12𝑧4Φ=0. (15) The solution of (15) which fulfills the condition 𝑑𝜓/𝑑𝜂∣_𝜂=0=0 is given by [13] 𝐶𝜓=¹𝜂𝑀_𝜔/2,02𝜔𝜂², (16) where 𝑀 is confluent hypergeometric function.

𝜓 function can be expressed as the following power series [14]: 1𝜓=𝐶1Γ))()((1−𝜔/22𝜔^1/2exp−𝜔𝜂²×^∞_𝑛=0Γ(𝑛+(1/2)−(𝜔/2))𝑛!𝑛!2𝜔𝜂^2𝑛,0≤𝜂≤1, (17) where Γ is the gamma function.

By employing the first- and second-type Bessel functions, respectively, 𝐼, 𝐾, the solution of (13) can be written as: 𝜓=𝐶₃𝐼0,𝐵𝜂+𝐶₄𝐾0,𝐵𝜂,1≤𝜂≤𝜎. (18) Constant 𝐶₁, 𝐶₃, and 𝐶₄ are obtained by using the mentioned boundary conditions in (14).

The Nusselt number can be obtained by employing (19) Nu=2𝑟₀𝜕𝑇𝜕𝑟_{𝑓,𝑟=𝑟0}𝑇𝑟₀,𝑥−𝑇_𝑏(𝑥)⁻¹(𝜉)=−2𝜑𝜃(1,𝜉)−𝜃_𝑏(𝜉), (19) where 𝜑(𝜉) is the dimensionless heat flux at solid-fluid interface: 𝜑(𝜉)=−𝜕𝜃𝜕𝜂_𝑓,𝜂=1=−𝑑𝜃₁𝑑𝜂_𝑓,𝜂=1sin(𝐵𝜉)−𝑑𝜃₂𝑑𝜂_𝑓,𝜂=1cos(𝐵𝜉)=𝜑₁sin(𝐵𝜉)+𝜑₂(),𝜑cos𝐵𝜉₁=−𝑑𝜃₁𝑑𝜂_𝑓,𝜂=1,𝜑₂=−𝑑𝜃₂𝑑𝜂_𝑓,𝜂=1. (20) And 𝜃_𝑏(𝜉) is the dimensionless bulk temperature: 𝜃_𝑏(𝜉)=4¹₀𝜃𝜂,𝜉𝜂1−𝜂²𝑑𝜂. (21) By using the above equations, the Nusselt number is obtained as (𝜉)2𝜑Nu=−¹sin(𝐵𝜉)+𝜑₂cos(𝐵𝜉)𝜃₁(1)−𝜃_1𝑏sin(𝐵𝜉)+𝜃₂(1)−𝜃_2𝑏(),cos𝐵𝜉 (22) where 𝜃_1𝑏,𝜃_2𝑏 are, respectively, the real and imaginary part of following integral: 𝜓_𝑏=4¹₀𝜓𝜂𝜂1−𝜂²𝑑𝜂. (23) One-time integration from (12) yields, 𝜓_𝑏=−4𝑖Pe𝐵𝑑𝜓𝑑𝜂_𝑓,𝜂=1=4𝑖𝜑Pe𝐵¹+𝑖𝜑₂. (24) On account of (24) the Nusselt number can be expressed as (𝜉)=𝜑Nu−2𝐵Pe₁sin(𝐵𝜉)+𝜑₂cos(𝐵𝜉)𝐵Pe𝜃₁(1)+4𝜑₂sin(𝐵𝜉)+𝐵Pe𝜃₂(1)−4𝜑₁().cos𝐵𝜉 (25) The average Nusselt number in an axial period is given by 𝐵Nu=2𝜋₀^2𝜋/𝐵𝑁(𝜉)𝑑𝜉. (26) Above integral has a singularity that is shown in (27): 𝜉_𝑛=tan⁻¹𝜃₂(1)−4𝜑₁/(𝐵Pe)𝜃₁(1)+4𝜑₂/()+𝐵Pe𝑛𝜋𝐵. (27) By employing the complex integral analysis and Residual theorem, the integral (26) is calculated as Nu=−8𝜑₁𝐵Pe𝜃₁(1)+4𝜑₂+8𝜑₂𝐵Pe𝜃₂(1)−4𝜑₁𝐵Pe𝜃₁(1)+4𝜑₂²+𝐵Pe𝜃₂(1)−4𝜑₁². (28) The simplified form of (28) is given by Nu=−8𝜑₁𝐵Pe𝜃₁(1)+8𝜑₂𝐵Pe𝜃₂(2)𝐵Pe𝜃₁(1)+4𝜑₂²+𝐵Pe𝜃₂(1)−4𝜑₁². (29) And finally Nu can be rewritten as Nu=4𝐵Pe𝐵Pe𝜓(1)−4𝑖(𝜑₁+𝜑₂)²𝑑𝜓²𝑑𝜂_𝑓,𝜂=1. (30)

4. Result and discussion

In Figure 2 three-dimensional plot of dimensionless temperature distribution versus 𝜉 and 𝜂 is shown, for 𝜎=1.15,𝛾=1.5,Pe=100, and 𝐵=1. Figure 2 shows by moving from wall to the center of duct the oscillation amplitude of temperature decreases, but in the center of duct the oscillation amplitude is not zero and Figure 3 confirms it. In this figure the dimensionless temperature versus 𝜉, for 𝜎=1.15,𝛾=1.5,Pe=100, and 𝐵=0.1 that is displayed. In Figure 3 it can be observed by closing to the center of duct not only the oscillation amplitude decreases, but also the sensing of wall temperature alteration is done with more longitudinal delay. In Figures 4 and 5 The dimensionless temperature distribution at the center of duct is reported for 𝜎=1.15,𝛾=1.5,andPe=100 and various values of 𝐵, Pe, respectively. The Figure 4 shows that increasing in 𝐵 causes a decrease in temperature oscillation amplitude in flow field.

Figure 2 

3D plot of dimensionless temperature distribution versus 𝜉 and 𝜂.

Figure 3 

Dimensionless temperature distribution at 𝜂=0,1,𝜎 versus 𝜉.

Figure 4 

Dimensionless temperature distribution at 𝜂=0 for 𝐵 = 0.3, 0.6, and 0.8 versus 𝜉.

Figure 5 

Dimensionless temperature distribution at 𝜂=0 for Pe = 20, 50, 100, and 200 versus 𝜉.

The reason of this phenomenon is related to the effect of conduction accompanied by convection heat transfer. Because of conduction heat transfer in Y direction, each point of fluid senses wall temperature alteration, immediately and without any longitudinal delay. That means by only taking into account conduction, when the wall temperature is Max at an axial position, for example X₀, the temperature of all points in fluid with same axial position as X₀, is Max, but with minus gradient in Y direction. About convection heat transfer, it should be mentioned that convection phenomenon transfers the upstream pointes property to downstream pointes in X direction; as a result of only taking into account convection, when the wall temperature is Max at an axial position, as X₀, the points with X₀ axial position are affected by upstream point only that has lower temperature in comparison to Max temperature. By increasing in 𝐵, frequency of oscillation increases, and it causes more difference between Max temperature; and upstream point temperature, as a result, contrast between convection and conduction increases and amplitude decreases.

In Figure 5 it can be observed that Pe number and 𝐵 have a same effect on oscillation amplitude of dimensionless temperature. According to Figure 5 an increase in Pe number causes an amplitude reduction. The reason of this behavior is also related to the difference between conducted and conduction phenomenon. Since Pe number is convection to conduction heat transfer rate ratio, increase in Pe number leads to increase in convection against conduction heat transfer as a result the temperature distribution in the fluid phase is more affected by upstream condition in comparison to wall boundary condition. In Figure 6 the dimensionless temperature of solid and fluid phases interface against 𝜉 for different values of 𝜎 is displayed.

Figure 6 

Dimensionless temperature distribution at solid-fluid interface for 𝜎=1.05,1.15,1.25versus 𝜉.

Figure 6 shows since increasing in wall thickness is followed with a decrease in conduction heat transfer, the oscillation amplitude in flow field decreases.

The longitudinal dimensionless temperature distribution is observed in Figure 7 for several values of 𝛾. In the Figure 7 it is seen that increase in heat conductivity of solid phase is followed with increase in oscillation amplitude, because of lower temperature gradient.

Figure 7 

Dimensionless temperature distribution at solid-fluid interface for 𝛾=0.5,1.5,3.0 versus 𝜉.

In Table 1, the values of the mean Nusselt number, evaluated through (30), are reported versus 𝐵.Pe and Pe. The table shows that the mean Nusselt number is an increasing monotonic function of 𝐵. Since the temperature gradient in Y direction has a direct relation with 𝐵, as a result larger value of 𝐵 leads to large value of Nusselt number.

Table 1 

The values of the mean Nusselt number for 𝜎=1.15,𝛾=1.5.

5. Conclusion

In the present paper, the heat transfer problem has been studied for the laminar forced convection of a fluid in a circular duct. The duct walls have finite width, and the temperature on the external boundaries varies longitudinally with sinusoidal law. The local energy balance equation has been solved analytically, with reference to the thermally developed region. Under this assumption, the temperature distribution is a periodic function of the axial direction, and the period is equal to the period of the wall temperature distribution and the following result is obtained.(i)The oscillation amplitude of the temperature field decreases by increasing the dimensionless frequency B and Peclet number.(ii)Increasing in The dimensionless frequency B is followed with increase an average Nusselt number.

Nomenclatures

𝐵∶	Dimensionless frequency
𝐶1,𝐶2,𝐶3,𝐶4:	Constants coefficient
𝑖:	Complex unit
𝑘:	Thermal conductivity
𝑛:	Positive integer
𝑀:	Confluent hypergeometricfunction
Nu:	Local Nusselt number
Nu:	Mean Nusselt number
𝑟:	Radial coordinate
𝑟0:	Internal radios
𝑟1:	External radios
𝑇:	Temperature
𝑢:	Fluid velocity
𝑢:	Mean fluid velocity
𝑥:	Axial coordinate.

Greek Symbols

𝛼:	Thermal diffusivity
𝛽:	Frequency
𝛾:	Dimensionless parameter, defined in  (5)
Δ𝑇:	Oscillation temperature of external wall
𝜉:	Dimensionless axial coordinate
𝜂:	Dimensionless radial coordinate
𝜃:	Dimensionless temperature
𝜃0,𝜃1,𝜃2:	Dimensionless function, defined in  (8)
Φ:	Dimensionless function, defined in (15)
𝜑1,𝜑2:	Dimensionless function, defined in (21)
𝜎:	Dimension radios of duct
𝜓:	Dimensionless complex function, defined in  (11).

Subscripts

𝑏:	Bulk quantity
𝑓:	Fluid
𝑠:	Solid.