Wall heat recirculation and exergy preservation in flow through a small tube with thin heat source

International Communications in Heat and Mass Transfer 64 (2015) 1–6

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International Communications in Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ichmt

Wall heat recirculation and exergy preservation in flow through a small tube with thin heat source☆ S.K. Som ⁎, Uttam Rana Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur 721302, India

a r t i c l e

i n f o

Available online 16 March 2015 Keywords: Heat recirculation Thin heat source Exergy balance

a b s t r a c t Theoretical studies have been made on heat transfer and exergy analysis of flow through a narrow tube with heat recirculating wall and embedded thin heat source. Heat transfer analysis is based on numerical solution of conservation equations of mass, momentum and energy, while the exergy analysis is based on flow exergy balance and entropy transport equation. It has been observed that the ratio of heat recirculation to heat loss (QR/QL) increases with increase in the ratio of thermal conductivity of solid wall to that of working fluid (ks/kg) and Peclet number of flow (Pe), while it decreases with an increase in external Nusselt number (NuE). The ratio QR/QL has a maximum with the ratio of wall thickness to tube radius (tw/R). The optimum value of tw/R depends only on ks/kg and reduces from a value of 0.2 at ks/kg = 330 to a value of 0.125 at ks/kg = 850, and then remains almost constant for any further increase in ks/kg. The volumetric entropy generation rate in the fluid flow reaches a maximum at a radial location close to the inner surface of the wall, while the entropy generation in solid wall, being more than that of the fluid, is radially uniform. The volumetric entropy generation is found to be confined within the upstream region of the heat source. Second law efficiency increases with a decrease in tw/R and NuE, but with an increase in Pe. It remains almost constant with ks/kg. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction The rapid growth of microdevices puts thrust on the development of micro and mesoscale thermal systems including combustors and power sources. This brings about technical challenges in designing such systems because of their small length scale which adds additional complex features in the process performed by the system. The scientific issues and challenges pertaining to micro and mesoscale power generating devices in practice have been addressed in recent reviews [1–3]. In small scale thermal systems with internal heat generation, the ratio of heat loss to heat generation becomes significant due to high surface area to volume ratio. This poses great problem in effective utilization of energy generated within the system for its desired performance. Amongst different approaches for thermal management in the performance of small scale systems, heat recirculation through conducting wall has gained considerable interest due to its natural occurrence in small structures. In the wake of growing concern for efficient utilization of natural energy resources, a system should be thermodynamically efficient in preserving the energy quality while meeting up the required performance criteria. The importance of exergy based thermodynamic analysis for the performance evaluation of thermal systems has been ☆ Communicated by W.J. Minkowycz. ⁎ Corresponding author. E-mail address: [email protected] (S.K. Som).

http://dx.doi.org/10.1016/j.icheatmasstransfer.2015.03.001 0735-1933/© 2015 Elsevier Ltd. All rights reserved.

reported in literature [4–10]. The destruction of exergy (the quality of energy) takes place due to the generation of entropy in a natural process. The methods of thermodynamic optimization in minimizing the irreversible production of entropy in a process for improving its efficiency pertaining to thermal systems have been addressed in the form of review work in literature [11–14]. Hardly any work is found, in the context of wall heat recirculation and entropy generation in internal flows through very narrow tube with embedded heat source, which may provide primary information about the optimum operating conditions for efficient use of energy source in the operation of small scale devices. In the present paper, an analysis of conjugate forced convection heat transfer in a narrow cylindrical tube with embedded heat source has been made. An exergy analysis has also been supplemented with the heat transfer analysis. The objective of the present work is to have a primary understanding of the physical process of heat transfer and thermodynamic irreversibility in the tube and to explore the optimum operating conditions in terms of wall thickness and wall thermal conductivity for maximum heat recirculation as well as exergy preservation in consideration of heat and exergy loss from outer wall of the tube. 2. Theoretical formulation Fig. 1(a). shows the schematic of the physical model consisting of a tube of radius R and length L with an embedded heat source at the middle of the tube. The heat source with a uniform volumetric energy generation

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S.K. Som, U. Rana / International Communications in Heat and Mass Transfer 64 (2015) 1–60

Nomenclature




u



Availability (W) A a Specific flow availability (J/kg) Ė Rate of entropy generation (W/K) 3 ė  Rate of volumetric entropy generation (W/m ·K) 3
iD Non-dimensional rate of volumetric entropy e ¼ eQT gen generation h Specific enthalpy (J/kg) k Thermal conductivity (W/m·K) ṁ Mass flow rate of fluid (kg/s) NuE External Nusselt number Pe Peclet number QR Heat recirculation through wall (W) QL Heat loss from outer wall of tube (W) QR⁎ Non-dimensional heat recirculation (QR/Qgen) Non-dimensional heat loss (QL/Qgen) Q⁎L Qgen(=πṡoR2Lc) Heat generation (W) r Dimensional radial coordinate Re Reynolds number s Specific entropy (J/kg·K) ṡ0 Volumetric energy generation rate (W/m3) u Axial velocity component (m/s) U Fluid velocity at the inlet of tube (m/s) v Radial velocity component (m/s) z Dimensional axial coordinate (m). 

" #


 ∂v
∂p
2 ∂2 v
1 ∂
∂v
∂v þ v ¼ − þ þ r ∂z
∂r
∂r
Re ∂z
2 r
∂r
∂r


" 2 2 ∂ θg 1 ∂ u þv ¼ þ ∂z
∂r
Pe ∂z
2 r
∂r



∂θg




∂θg

∂θg

!#

∂r



 þf z

ð4Þ



Greek symbols α Thermal diffusivity of fluid (m2/s) ρ Density of fluid (kg/m3) θ Non-dimensional temperature ηII Second law efficiency. Subscript g i o s r

where



 f z ¼1

for

¼0







ð5Þ

The non-dimensional variables used in the Eqs. (1)–(5) are given by 2RU i ; Re ¼ 2RU z
¼ Rz ; r
¼ Rr ; u
¼ Uu ; v
¼ Uv ; p
¼ ρUp 2 ; θg;s ¼ T T−T ν ; Pe ¼ α ; re f −T i

L
¼ RL ; L
c ¼ LRc. The reference temperature Tref, used in the normalization of temperature T, is defined in relation to volumetric heat source ṡ0 as 

T re f ¼ T i þ

s0 Lc : ρUcp

ð6Þ

The physical significance of Tref implies a fluid temperature that can be raised adiabatically from a temperature of Ti by the utilization of total energy generated from the volumetric heat sources (ṡ0). The boundary conditions for the solution of Eqs. (1)–(5) are as

follows:u* = 1; v* = 0; θg = 0 at z⁎ = 0 and 0 ≤ r* ≤ 1∂u
¼ 0; ∂v
¼ ∂θg ∂z


∂z

∂z

¼ 0 at z* = L* and 0 ≤ r* ≤ 1u*(1, z*) = 0; v*(1, z*) = 0; ∂θ




at r* = 0 and 0 ≤ z* ≤ L*




∂θ

s w ¼ Nu2E θw at r
¼ Rþt and − kkgs ∂θ R ∂r


0 ≤ z* ≤ L*where NuE ¼ 2hk∞g R and h∞ is external convective heat transfer coefficient. A finite volume method is used to discretize the continuity, momentum and energy equations in gas phase and energy equation in a solid phase. The conservation equations along with the boundary conditions are solved in a segregated manner. The segregated solver first solves the momentum equations, then solves the continuity equation, and update the pressure and velocity. The energy equations are subsequently solved to desired level of accuracy. Grid independent test is carried out to ensure the numerical results are independent of grid size. Finer grids are used near the wall and source where gradients are prominent.

(ṡ0) has been considered to be confined over the entire cross section but within a length of Lc. The radius of the tube and the value of Lc are considered to be 0.5 mm. The length of the tube is taken to be L = 20R. It can be mentioned in this context that the dimensions of the tube are chosen so that it fall in the range of typical microcombustors [2]. Air is considered to be the fluid flowing through the tube. The flow is assumed to be steady, Newtonian, incompressible and axisymmetric. Such a small system with flow of air belong to the domain of continuum with no slip at the solid wall, but exhibits a very high ratio of heat loss to heat generation because of high ratio of surface area to volume. The conservation equations of mass, momentum and energy in dimensionless form are written in a cylindrical polar coordinate system as

2.1. Exergy analysis The analysis is based on an exergy balance to a control volume as shown in Fig. 1(b). Therefore it can be written 









A f ;o ¼ A f ;i þ As −AL − I :

ð7Þ



A f ;o , the rate of flow exergy out of the tube = ṁaf,o where ṁ is mass flow rate of air. af,o, the specific flow exergy = hf,o − Trsf,o where h f ;o ¼ hr T f ;o

T f ;o

þ ∫ cp ðT ÞdT; s f ;o ¼ sr þ ∫

" #



2

 ∂u ∂p 2 ∂ u 1 ∂
∂u
∂u þ v ¼ − þ þ r u ∂z
∂r
∂z
Re ∂z
2 r
∂r
∂r





s θg = θs; kg ∂r
g ¼ ks ∂θ at r* = 1 and 0 ≤ z* ≤ L* ∂u ¼ 0; ∂v ¼ 0; ∂r
g ¼ 0 ∂r
∂r
∂r


gas inlet outlet solid exergy reference state

∂u
1 ∂ðr
v
Þ þ ¼0 ∂z
r
∂r





L −Lc L þ Lc
≤z ≤ 2 2 otherwise


 ∂2 θs 1 ∂
∂θs þ

r ¼ 0:

2 r ∂r ∂r ∂z

0;

Superscript * Non-dimensional variables




r




ð3Þ

Tr

ð1Þ

Tr 

AL is the availability loss associated with heat loss from outer wall of the tube, and is given by
 ZL   ∂T  T AL ¼ 2πðR þ t w Þ ks  s  1− r dz: Tw ∂r r¼t w þR 

ð2Þ

cp ðT ÞdT : T

0

S.K. Som, U. Rana / International Communications in Heat and Mass Transfer 64 (2015) 1–6

NuE,Ti

3

L tw

U , Ti

R

r z

Axisymmetry Lc

(a) AL

A f ,i

control surface

A f ,0

As

(b) Fig. 1. (a). The schematic of a microtube with embedded heat source. (b) The control volume for exergy analysis of the present model.

İ, thermodynamic irreversibility occurring within the control volume can be written as 



I¼ T r E : The rate of entropy generation (Ė) in the present simulation is due to heat diffusion and viscous dissipation in the flow filed. However, the entropy generation due to viscous dissipation is neglected since it is relatively much smaller compared to that due to heat diffusion as reported earlier [15]. This is because of relatively small flow velocities and their gradients. The volumetric entropy generation rate in fluid flow and solid wall can be written as 

eg ¼

kg

"

∂T g ∂z

T 2g

!2 þ

∂T g

!2 # 

; es ¼

∂r

ks T 2s

"

2 # ∂T s 2 ∂T s þ : ∂z ∂r

The total entropy generation rate in both the phases are determined as



ZL ZR

ZL Rþt Zw 

E¼ 2π



eg rdrdz þ 2π 0

0

es rdrdz: 0

R

The flow availability has been evaluated with respect to an exergy reference thermodynamic state of Pr = 101.34 kN/m2, Tr = 298.15 K. The pressure and temperature of fluid at inlet are assumed to be same as those of exergy reference state. This leads to zero flow exergy at 



inlet of the control volume (A f ;i ). Eq. (7) is used to determine As , the rate of exergy addition due to embedded heat source. The second law efficiency ηII is evaluated as 

ηII ¼

A f ;o 

As

:

3. Results and discussions The variations of dimensionless heat recirculation (QR⁎) and heat loss (Q⁎L ) with the ratio of wall thickness to tube radius (tw/R) are depicted in Fig. 2(a). For a given value of ks/kg and Pe, both heat recirculation and heat loss increase with tw/R. However, the increase in (QR⁎) is sharp at lower values of tw/R and becomes marginal thereafter. An increase in external Nusselt number NuE results in an increase in Q⁎L but a decrease in QR⁎. Increase in Q⁎L with tw/R is due to an increase in outer surface area of the tube, since the value of outer radius is well below the critical radius of insulation under the present situation. An increase in Q⁎ L with tw/R must be supplemented by an increase in radial conduction through the wall. This results in a higher wall temperature and an increase in axial conduction in the solid wall. This in turn results in an increase in QR⁎. Fig. 2(b) shows the variation of QR/QL with tw/R for different values of external Nusselt number NuE. It is observed that QR/QL increases with tw/R, and then reaching a maximum, decreases thereafter. This physically implies that in the lower range of tw/R, the increase in QR dominates over that in QL, while the trend is reversed thereafter. The value of tw/R at which QR/QL reaches a maximum (Fig. 2b) is 0.2 (for ks/kg = 330 and Pe = 24), which is independent of NuE. Fig. 3(a). shows that an increase in the ratio of thermal conductivity of solid wall to that of gas ks/kg results in an increase in QR/QL at any value of tw/R. This can be physically explained due to the fact that an increase in wall thermal conductivity ks enhances the rate of axial conduction through wall. This results in a higher and almost axially uniform temperature of inner wall surface which enhances the heat recirculation. The interesting feature observed in this context is that the optimum value of tw/R for (QR/QL)max decreases from a value of 0.2 at ks/kg = 330 to a value of 0.125 at ks/kg = 850 (Fig. 3b), while it remains almost constant with any further increase in ks/kg. It is further observed (Fig. 3b) that the increase in the maximum value of QR/QL with ks/kg is sharp at lower values of ks/kg, but it becomes relatively marginal with further increase in ks/kg beyond 880. It has also been observed that the ratio QR/QL increases with Peclet number Pe, while the optimum value of tw/R for maximum QR/QL is independent of Pe.

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S.K. Som, U. Rana / International Communications in Heat and Mass Transfer 64 (2015) 1–6

(a)

(a)

(b) (b) Fig. 2. (a). Variation of heat loss, heat recirculation and (b) variation of the ratio of heat loss to heat recirculation with tw/R for different values of NuE.

The radial distribution of non-dimensional volumetric entropy generation rate in the fluid and the same in the solid wall at different axial locations are shown in Fig. 4a and b respectively for a given set of operating conditions (ks/kg = 330, tw/R = 0.2, NuE = 0.4, Pe = 24). It is observed (Fig. 4a) that the maximum entropy generation in the fluid flow takes place at a radial location near the wall. The entropy generation is attributed to radial and axial temperature gradients and the local temperature. At any section, the entropy generation distribution shows a maximum at a radial location near the inner surface of the wall due to the combined effect of high radial and axial temperature gradients, while it is negligible near the axis where both the temperature gradients are very small. At a section close to the heat source (z⁎ = 9.5) (Fig. 4a), the entropy generation is relatively high as compared to that of other sections and the radial distribution becomes more peaky. This is attributed to the predominant axial temperature gradient compared to the radial one near the heat source. The entropy generation at any radial location becomes very low at a section downstream of the heat source (z⁎ = 15). Volumetric entropy generation in solid wall shows (Fig. 4b) almost a uniform radial distribution except a decreasing trend very close to the inner surface of the wall. The entropy generation is high at a section close to the heat source, while it is almost negligible at a section downstream of the source. It is important

Fig. 3. (a) Variation of heat loss to heat recirculation ratio with tw/R for different values ks/ kg. (b) Variation of optimum wall thickness ratio with ks/kg. (Pe = 24, NuE = 0.4).

to mention in this context that though the entropy generation in solid wall is higher than that in the flowing fluid, the enhancement is much smaller as compared to the ratio of thermal conductivity ks/kg which, in the present situation, is 330. This is attributed to very small temperature gradients in the solid wall as compared to those in the fluid. Fig. 5a–b shows the variations of second law efficiency with tw/R for different values of NuE and Pe. It is observed that ηII decreases monotonically with tw/R. A change in second law efficiency is intrinsically related to a change in heat loss and heat recirculation characteristics. It has already been observed that both heat loss QL and heat recirculation QR increases with tw/R. While the increase in QL increases the availability loss, the increase in QR increases the irreversibility due to enhanced heat conduction. The combined effect of the two results in a decrease in ηII. A decrease in NuE or increase in Pe decreases the heat loss (QL) 

and hence in the associated exergy loss (AL ). This results in an increase in ηII (Fig. 5a–b). It has been observed that ηII is almost independent of ks/kg. Therefore, it is observed from the present study that a value of tw/R equals to 0.125 and a thermal conductivity ratio close 850 correspond to a high and energy efficient heat recirculation process. However, a trade-off between heat recirculation QR and second law efficiency ηII may be made with a choice of tw/R below its optimum value based on an overall economy of the desired process in actual system.

S.K. Som, U. Rana / International Communications in Heat and Mass Transfer 64 (2015) 1–6

5

(a) (a)

(b) (b) Fig. 4. Radial distribution of volumetric entropy generation rate at different axial locations of the tube in (a) fluid and (b) solid wall.

4. Conclusion A theoretical study on heat transfer along with transfer and destruction of exergy has been made for a flow through a small tube with heat recirculating wall and embedded thin heat source. The major observations of the study are as follows: • The ratio of heat recirculation to heat loss QR/QL increases with an increase in the ratio of thermal conductivity of solid wall to that of fluid ks/kg and a decrease in external Nusselt number NuE. • The ratio QR/QL increases with the ratio of wall thickness to tube radius tw/R and reaches a maximum and then decreases thereafter with tw/R. The optimum value of tw/R depends only on the thermal conductivity ratio ks/kg and is independent of NuE and Pe. With an increase in ks/kg from 330 to 850, the optimum value of tw/R decreases from 0.2 to 0.125, while it remains almost constant with any further increase in ks/kg. • The value of (QR/QL)max increases sharply with ks/kg for values of ks/kg below 850, while it changes marginally with an increase in ks/kg beyond 850. • The entropy generation is confined in the upstream region of heat source. The volumetric entropy generation rate in fluid flow reaches

Fig. 5. Variation of second law efficiency with wall thickness ratio (tw/R) for different values of (a) external Nusselt number NuE and (b) Peclet number Pe.

a maximum at a radial location close to the inner surface of the wall. The entropy generation in solid wall, being more than that of the fluid, is radially uniform. • Second law efficiency ηII increases with a decrease in tw/R and NuE, but with an increase in Pe, while it remains almost constant with ks/kg. • A tradeoff between heat recirculation and second law efficiency may be made with a choice of tw/R below its optimum value leading to an overall economy of an actual system in practice.

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