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Dimensioning of ducts for maximal volumetric heat transfer taking both laminar and turbulent flow possibilities into consideration

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Abstract

It is intended to design compact heat exchangers which can transfer high heat flow for a given volume and temperature difference with high efficiency. This work presents the optimal design of heat exchangers for a given length or hydraulic diameter with a constraint of a certain pressure loss and constant wall temperature. Both volumetric heat transfer and heat transfer efficiency are taken into consideration for the design in laminar or turbulent flow regions. Equations are derived which easily enable optimal design for all shapes of ducts and for all Pr numbers. It is found that optimum conditions for turbulent flow is possible for all duct hydraulic diameters; however, it is possible to have optimum conditions till a certain dimensionless duct hydraulic diameter for laminar flow. Besides maximal volumetric heat flow, heat transfer efficiency should be taken into consideration in turbulent flow for optimum design.

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Abbreviations

RMSE :

Root mean square error

A :

Cross-sectional area

c p :

Specific heat at constant pressure

d h :

Hydraulic diameter, Eq. (15)

ET :

Equilateral triangular

h :

Heat transfer coefficient

k :

Thermal conductivity

K :

Incremental pressure loss coefficient, Table 1

L :

Length of the duct

L * :

Dimensionless length of the duct, Eq. (10)

Nu :

Nusselt number, Eq. (14)

P :

Pressure

Pr:

Prandtl number

\( \dot{Q} \) :

Heat flow

\( \dot{q}_{V} \) :

Volumetric heat flow

PP :

Parallel plate

Re:

Reynolds number, Eq. (19)

t :

A dimensionless quantity, Eq. (34)

T :

Temperature

u :

Velocity

u p :

Pressure velocity, Eq. (6)

V :

Volume

x * :

Dimensionless duct length, Eq. (24)

y :

A dimensionless quantity, Eq. (36)

δ :

Length defined according to Eq. (11)

Δ:

Difference

η :

Efficiency, Eqs. (47) and (48)

φ :

Friction shape factor in laminar duct flow

λ :

Pressure loss coefficient

ν :

Kinematic viscosity

ρ :

Density

θ :

Dimensionless temperature, Eq. (7)

cr :

Critical

e :

Exit

f :

Frictional

i :

Inlet

max:

Maximum

o :

Optimum

V :

Volumetric

w :

Wall

*:

Dimensionless

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Author information

Authors and Affiliations

  1. Department of Automotive Engineering, Çukurova University, 01330, Adana, Turkey

    Alper Yilmaz

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  1. Alper Yilmaz
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Correspondence to Alper Yilmaz.

Appendix

Appendix

Two examples are given for the better understanding of the optimum design problem.

1.1 Example #1

Air is used as heat transfer medium at constant wall temperature. Parallel plate duct of 7.5 mm gap is used. Allowed pressure loss is 100 Pa. Physical properties of air are assumed as constants as follows (at 1 bar, 20 °C):

$$ \rho = 1.1881:c_{p} = 1007:\nu = 1.535\;10^{ - 5} :Pr = 0.7149 $$

The results for optimal design are presented in Table 6.

Table 6 Results for the Examples 1 and 2 in the “Appendix” (dimensions are in SI)
Full size table

In this example, optimum condition in laminar flow is not possible, because \( d_{h,o,cr}^{*} \) = 4,611 and it is less than real \( d_{h,o}^{*} \) value of 12,674 calculated according to Eq. (18). This can be seen from Re = 9,066, also. Therefore, only turbulent flow condition is considered. At optimum condition, length of the PP duct is 0.859 meters, velocity is 9.278 m/s and volumetric heat flow is \( \dot{q}_{V,o} \) = 8,266 W/m3 °C. The efficiency (\( \eta \) = 0.6397) at the optimum point can be considered as good. Therefore, optimum design parameters are determined.

1.2 Example #2

In this example, channel is ET duct with a side length of 3 mm. The allowed pressure loss is 200 Pa. Physical properties of air which is the heat transferring medium are the same with the values in the previous example #1. The results for optimal design are presented in Table 6.

In this example, optimum conditions for both laminar and turbulent flow regimes are possible, because \( d_{h,o,cr}^{*} \) = 4,824 which is greater than the real value \( d_{h,o}^{*} \) = 2,070.5 which is determined according to Eq. (18). Therefore, both cases should be analyzed.

Duct length is 96.2 mm in laminar flow case, whereas it is only 13.7 mm in turbulent flow case. The velocities are found as 8.765 and 28.90 m/s, respectively. \( \dot{q}_{V,o} \) value in turbulent flow is 418,463 W/m3 °C which is much higher than 75,680 W/m3 °C value of laminar flow. However, efficiency is only 16.59 % in turbulent flow, whereas, it is 69.45 % in laminar flow. In this example, laminar flow optimal condition can be preferred, because the efficiency in laminar flow is very high as compared to the efficiency in turbulent flow.

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Yilmaz, A. Dimensioning of ducts for maximal volumetric heat transfer taking both laminar and turbulent flow possibilities into consideration. Heat Mass Transfer 51, 543–552 (2015). https://doi.org/10.1007/s00231-014-1432-z

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