Nusselt number and friction factor correlation development for arc-shape apex upstream artificial roughness in solar air heater

Abstract

In the present work, an outdoor experimental investigation for solar air heater with arc-shape apex upstream flow by the use of circular cross-sectional wires as roughness elements has been carried out. The roughness elements have been expressed in non-dimensionalizing geometric parameters as relative roughness pitch (P/e), relative roughness height (e/D), and flow attack angle (α/60), and the range of these parameters varies from 8 to 15, 0.0454, and 0.75 to 1.25, respectively. For evaluation of performance of the roughened SAH, a novel parameter has been proposed and introduced in the present investigation which is thermo-hydraulic improvement parameter (THIP). With the use of present roughness geometry, considerably, Nusselt number enhancement ratio (NNER) and friction factor enhancement ratio (FFER) have been observed. The maximum NNER and FFER values obtained experimentally are about 2.83 and 1.79 times, respectively, while the maximum THIP obtained is 157.49% higher than the smooth SAH. Using the experimental results, correlations for the output parameters (Nusselt number and friction factor) as a function of input parameters (flow and roughness) have been developed.

Appendices

Appendix 1

Various performance parameters and improvement factors.

Table 4, 5, 6, 7, 8, 9

Table 4 Various performance parameters and improvement factors for arc-shape rib SAH (Saini and Saini (2008))

Table 5 Various performance parameters and improvement factors for V-shape rib SAH (Momin et al. (2002))

Table 6 Various performance parameters and improvement factors for W-shape rib SAH (Lanjewar et al.(2011a, b))

Table 7 Various performance parameters and improvement factors for inclined continuous rib SAH (Gupta et al. (1997))

Table 8 Various performance parameters and improvement factors for discrete W-shape rib SAH (Kumar et al.(2009))

Table 9 Various performance parameters and improvement factors for the present study

Appendix 2

Uncertainty analysis.

A methodology for evaluation of the uncertainty in experimental results which has been suggested by Kline and McClintock (1953) is used in the present work. The procedure is as below:

If a parameter is calculated using certain measured quantities as,

$$Y=y\left({x}_{1},{x}_{2},{x}_{3}\cdots \cdots \cdots \cdots {x}_{n}\right)$$

Then, uncertainty in the measurement of “y” is given as follows:

$$\frac{\delta y}{y} = \left[ {\left( {\frac{\partial y}{{\partial x_{1} }}\delta x_{1} } \right)^{2} + \left( {\frac{\partial y}{{\partial x_{2} }}\delta x_{2} } \right)^{2} + \left( {\frac{\partial y}{{\partial x_{3} }}\delta x_{3} } \right)^{2} + ......\left( {\frac{\partial y}{{\partial x_{n} }}\delta x_{n} } \right)^{2} } \right]^{0.5}$$

• (1) Area of absorber plate (A_c):

  $$\begin{array}{*{20}c} {A_{c} = W \times L} \\ {\frac{{\delta A_{c} }}{{A_{c} }} = \left[ {\left( {\frac{\delta L}{L}} \right)^{2} + \left( {\frac{\delta W}{W}} \right)^{2} } \right]^{0.5} } \\ \end{array}$$

• (2) Cross-sectional area of air flow duct (A)

  $$\begin{array}{*{20}c} {A = W \times H} \\ {\frac{\delta A}{A} = \left[ {\left( {\frac{\delta W}{W}} \right)^{2} + \left( {\frac{\delta H}{H}} \right)^{2} } \right]^{0.5} } \\ \end{array}$$

• (3) Perimeter of duct (P)

  $$\begin{array}{*{20}c} {P = 2(W + H)} \\ {\frac{\delta P}{P} = \left[ {\left( {2.\frac{\delta W}{W}} \right)^{2} + \left( {2\frac{\delta H}{H}} \right)^{2} } \right]^{0.5} } \\ \end{array}$$

• (4) Hydraulic diameter of duct (D_h)

  $$\begin{array}{*{20}c} {D_{h} = \frac{2.WH}{{\left( {W + H} \right)}}} \\ {\frac{{\delta D_{h} }}{{D_{h} }} = \left[ {\left( {\frac{\delta A}{A}} \right)^{2} + \left( {\frac{\delta P}{P}} \right)^{2} } \right]^{0.5} } \\ \end{array}$$

• (5) Area of orifice meter (A_o)

  $$\begin{array}{*{20}c} {A_{o} = \frac{\pi }{4} \times D_{2}^{2} } \\ {\frac{{\delta A_{o} }}{{A_{o} }} = \left[ {\frac{{2\delta D_{2} }}{{D_{2} }}} \right]} \\ \end{array}$$

• (6) Density (ρ)

  $$\begin{array}{*{20}c} {\rho = \frac{{P_{atm} }}{{R.T_{fo} }}} \\ {\frac{\delta \rho }{\rho } = \left[ {\left( {\frac{{\delta P_{atm} }}{{P_{atm} }}} \right)^{2} + \left( {\frac{{\delta T_{o} }}{{T_{o} }}} \right)^{2} } \right]^{0.5} } \\ \end{array}$$

• (7) Mass flow rate (m)

  $$\begin{array}{*{20}c} {m = C_{d} .A_{o} \sqrt {\frac{{2\rho (\Delta P_{o} )}}{{1 - \beta_{R}^{4} }}} } \\ {\frac{\delta m}{m} = \left[ {\left( {\frac{{\delta C_{d} }}{{C_{d} }}} \right)^{2} + \left( {\frac{{\delta A_{o} }}{{A_{o} }}} \right)^{2} + \left( {2\frac{\delta \rho }{\rho }} \right)^{2} + \left( {\frac{{\delta (\Delta P_{o} )}}{{(\Delta P_{o} )}}} \right)^{2} \left( {\frac{{2.\beta_{R}^{3} .\delta \beta_{R} }}{{1 - \beta_{R}^{2} }}} \right)} \right]^{0.5} } \\ \end{array}$$

• (8) Reynolds number (Re)

  $$\begin{array}{*{20}c} {{\text{Re}} = \frac{{\rho VD_{h} }}{\mu }} \\ {\frac{{\delta {\text{Re}} }}{{\text{Re}}} = \left[ {\left( {\frac{\delta V}{V}} \right)^{2} + \left( {\frac{\delta \rho }{\rho }} \right)^{2} + \left( {\frac{{\delta D_{h} }}{{D_{h} }}} \right)^{2} + \left( {\frac{\delta \mu }{\mu }} \right)^{2} } \right]^{0.5} } \\ \end{array}$$

• (9) Useful heat gain (Qu)

  $$\begin{array}{*{20}c} {Qu = \dot{m}C_{pa} .\Delta T} \\ {\frac{\delta Qu}{{Qu}} = \left[ {\left( {\frac{\delta m}{m}} \right)^{2} + \left( {\frac{{\delta C_{pa} }}{{C_{pa} }}} \right)^{2} + \left( {\frac{\delta (\Delta T)}{{\Delta T}}} \right)^{2} } \right]^{0.5} } \\ \end{array}$$

• (10) (10). Heat transfer coefficient (h)

  $$\frac{\delta h}{h} = \left[ {\left( {\frac{\delta Qu}{{Qu}}} \right)^{2} + \left( {\frac{{\delta A_{c} }}{{A_{c} }}} \right)^{2} + \left( {\frac{{\delta (T_{pm} )}}{{(T_{pm} )}}} \right)^{2} } \right]^{0.5}$$

• (11) (11). Nusselt number (Nu)

  $$\frac{\delta Nu}{{Nu}} = \left[ {\left( {\frac{\delta h}{h}} \right)^{2} + \left( {\frac{{\delta D_{h} }}{{D_{h} }}} \right)^{2} + \left( {\frac{{\delta (K_{a} )}}{{(K_{a} )}}} \right)^{2} } \right]^{0.5}$$

• (12) Friction factor (f)