Experimental study of thermal augmentation in solar air heater roughened with aligned gaps in V-rib roughness with staggered element geometry

Abstract

The most suitable approach to augment convective heat release rate is to fabricate the roughness underside of solar air heater (SAH) duct. This paper reveals an experimental effort made to augment the thermal performance of aligned gaps along with staggered element on SAH with minimum frictional consequence. The V-rib roughness namely aligned gaps in V-rib with staggered element roughness is evaluated with investigated range of parameters as relative rib height (e/D_h) 0.043, number of gaps (N_g) 2–5, relative gap breadth (g/e) 2–5, relative rib pitch (P/e) 10, relative staggered roughness pitch (P’/P) 0.4 and relative staggered rib length (r/g) 1. As per the selected mass flow rate Reynolds number varied from 4000 to 14,000. The result shows the augmentation in Nusselt number (Nu) value and friction factor (f) which is recorded as 2.16 times and 2.73 times respectively compared to flat plate surface. Performance of aligned gaps in V-rib with staggered element corresponding to studied geometrical parameters of roughness is compared with contemporary better performing V- rib geometry has also been done.

Appendix 1: Uncertainty analysis

If the parameters are calculated using certain reflected quantities as:

$$y = y\left( {x_{1} ,x_{2} ,x_{3} ,\ldots, x_{n} } \right)$$

Then the uncertainty in ‘y’ is measured by:

$$\frac{\delta y}{y} = \sqrt {\left( {\frac{\delta y}{{\delta x_{1} }} \times \delta x_{1} } \right)^{2} + \left( {\frac{\delta y}{{\delta x_{2} }} \times \delta x_{2} } \right)^{2} + \left( {\frac{\delta y}{{\delta x_{3} }} \times \delta x_{3} } \right)^{2} + .......\left( {\frac{\delta y}{{\delta x_{n} }} \times \delta x_{n} } \right)^{2} }$$

where, δx₁, δx₂, δx₃ ….. δx_n are the uncertainties in measurement of x_1, x_2, x₃…. x_n, δy and $\frac{\delta y}{y}$ are absolute uncertainty and relative uncertainty respectively.

The important parameters in present investigation are:

• i. Reynolds Number, ${\text{Re}} = \frac{{\rho_{a} VD_{h} }}{\mu }$

• ii. Heat transfer coefficient,$h = \frac{{mC_{p} \Delta T}}{{A_{p} \left( {T_{P} - T_{m} } \right)}}$

• iii. Nusselt number, $Nu = \frac{{hD_{h} }}{k}$

• iv. Friction factor, $f = \frac{{D_{h} \left( {\Delta P_{d} } \right)}}{{2L_{f} V^{2} \rho }}$

Appendix Table 4 gives a detail of uncertainty interval of various measurements and equipment.

Table 4 Uncertainty interval of various measurements and equipment

Appendix Table 5 gives the data related to dimensional parameters taken during present study.

Table 5 Data related to dimensional parameters

Thermo-physical properties of air at T_f = 298.43, using correlations available are:

$$C_{p} = 1006 \times \left( {\frac{{T_{m} }}{293}} \right)^{0.0155} = { 1}00{6}.{\text{29 J}}/{\text{kg K}}$$

$$\mu = 1.81 \times 10^{ - 5} \times \left( {\frac{{T_{m} }}{293}} \right)^{0.735} = { 1}.{\text{834 x 1}}0^{{ - {5}}} {\text{Pa s}}$$

$$k = 0.0257 \times \left( {\frac{{T_{m} }}{293}} \right)^{0.86} = \, 0.0{261}0{\text{9 W}}/{\text{m K}}$$

• 1. Area of absorber plate (A_p):

  $$A_{p} = W \times L$$
  $$\frac{{\delta A_{p} }}{{A_{p} }} = \left[ {\left( {\frac{{\delta A_{p} }}{\delta L} \times \delta L} \right)^{2} + \left( {\frac{{\delta A_{p} }}{\delta W} \times \delta W} \right)^{2} } \right]^{1/2}$$
  $$\frac{{\delta A_{p} }}{{A_{p} }} = \left[ {\left( {\frac{\delta L}{L}} \right)^{2} + \left( {\frac{\delta W}{W}} \right)^{2} } \right]^{1/2} = \left[ {\left( \frac{1}{1000} \right)^{2} + \left( {\frac{0.1}{{300}}} \right)^{2} } \right]^{1/2} = \, 1.054 \times 10^{ - 3} = \, 0.1054\%$$

• 2. Area of flow (A):

  $$A = W \times H$$
  $$\delta A = \left[ {\left( {\frac{\delta A}{{\delta W}} \times \delta W} \right)^{2} + \left( {\frac{\delta A}{{\delta H}} \times \delta H} \right)^{2} } \right]^{1/2}$$
  $$\frac{\delta A}{A} = \left[ {\left( {\frac{\delta W}{W}} \right)^{2} + \left( {\frac{\delta H}{H}} \right)^{2} } \right]^{1/2} = \left[ {\left( {\frac{0.1}{{300}}} \right)^{2} + \left( {\frac{0.1}{{25}}} \right)^{2} } \right]^{1/2} = \, 4.013 \times 10^{ - 3} = \, 0.4013\%$$

• 3. Hydraulic diameter (D_h):

  $$D_{h} = \frac{4A}{P} = \frac{2WH}{{\left( {W + H} \right)}} = 2\left( {WH} \right)\left( {W + H} \right)^{ - 1}$$
  $$\frac{{\partial D_{h} }}{\partial H} = 2\left( {WH} \right)\left( { - 1} \right)\left( {W + H} \right)^{ - 2} + \left( {W + H} \right)^{ - 1} \left( {2W} \right)$$
  $$\begin{gathered} = \frac{ - 2WH}{{\left( {W + H} \right)^{2} }} + \frac{2W}{{W + H}}or\frac{2W}{{W + H}} - \frac{2WH}{{\left( {W + H} \right)^{2} }} = \frac{2 \times 300}{{\left( {300 + 25} \right)}} - \frac{2 \times 300 \times }{{\left( {300 + 25} \right)^{2} }} \hfill \\ = 1.84615 - 0.142012 = 1.704 \hfill \\ \end{gathered}$$
  $$= \frac{2 \times 25}{{\left( {300 + 25} \right)}} - \frac{2 \times 300 \times 25}{{\left( {300 + 25} \right)^{2} }} = 0.153846 - 0.142012 = 0.011834$$
  $$\frac{{\delta D_{h} }}{{D_{h} }} = \frac{{\left[ {\left( {\frac{{\delta D_{h} }}{\delta H}.\delta H} \right)^{2} + \left( {\frac{{\delta D_{h} }}{\delta W}.\delta W} \right)^{2} } \right]}}{{2\left( {WH} \right)\left( {W + H} \right)^{ - 1} }}^{1/2} = \frac{{\left[ {\left( {1.704 \times 0.1} \right)^{2} + \left( {0.011834 \times 0.1} \right)^{2} } \right]^{1/2} }}{{{{\left( {2 \times 300 \times 25} \right)} \mathord{\left/ {\vphantom {{\left( {2 \times 300 \times 25} \right)} {\left( {300 + 25} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {300 + 25} \right)}}}} = 3.69 \times 10^{ - 3} = 0.369\%$$

• 4. Area of orifice meter (A_o):

  $$A_{o} = \frac{\pi }{4}d_{o}^{2}$$
  $$\frac{{\delta A_{o} }}{{\delta d_{o} }} = \frac{{2\pi d_{o} }}{4}$$
  $$\delta A_{o} = \left[ {\left( {\frac{{\delta A_{o} }}{{\delta d_{o} }} \times \delta d_{o} } \right)^{2} } \right]^{1/2} = \left[ {\left( {\frac{\pi }{2}d_{o} \times \delta d_{o} } \right)^{2} } \right]^{1/2}$$
  $$\delta A_{o} = \left( {\frac{\pi }{2}d_{o} \times \delta d_{o} } \right)$$
  $$\frac{{\delta A_{o} }}{{A_{o} }} = \frac{{\pi /2 \cdot d_{o} \cdot \delta d_{o} }}{{\pi /4 \cdot d_{o} }} = \frac{{2 \cdot \delta d_{o} }}{{d_{o} }} = \frac{2 \times 0.1}{{40.5}} = 4.9382 \times 10^{ - 3} = 0.49382\%$$

• 5. Density of air (ρ_o):

  $${\text{Density of air at T}}_{{\text{o}}} {, }\rho_{o} = \frac{{P_{atm} }}{{RT_{o} }}$$
  $$\frac{{\delta \rho_{o} }}{{\rho_{o} }} = \left[ {\left( {\frac{{\delta P_{atm} RT_{o} }}{{RT_{o} P_{atm} }}} \right)^{2} + \left( {\frac{{ - P_{atm} \delta T_{o} RT_{o} }}{{\left( {RT_{o} } \right)^{2} P_{atm} }}} \right)^{2} } \right]^{1/2}$$
  $$\frac{{\delta \rho_{o} }}{{\rho_{o} }} = \left[ {\left( {\frac{{\delta P_{atm} }}{{P_{atm} }}} \right)^{2} + \left( {\frac{{ - \delta T_{o} }}{{RT_{o} }}} \right)^{2} } \right]^{1/2} = \frac{{\delta \rho_{o} }}{{\rho_{o} }} = \left[ {\left( \frac{1}{760} \right)^{2} + \left( {\frac{ - 0.25}{{300.91}}} \right)^{2} } \right]^{1/2} = \, 1.556 \times 10^{ - 3} {\text{or }}0.1556\%$$
  $${\text{Density of air at Tm, (}}\rho_{{\text{a}}} {):}$$
  $$\frac{{\delta \rho_{a} }}{{\rho_{a} }} = \left[ {\left( {\frac{{\delta P_{a} }}{{P_{a} }}} \right)^{2} + \left( {\frac{{ - \delta T_{m} }}{{T_{m} }}} \right)^{2} } \right]^{1/2} = \frac{{\delta \rho_{a} }}{{\rho_{a} }} = \left[ {\left( \frac{1}{760} \right)^{2} + \left( {\frac{ - 0.25}{{298.43}}} \right)^{2} } \right]^{1/2} = { 1}.{559} \times {1}0^{{ - {3}}} {\text{or }}0.{1559}\%$$

• 6. Mass flow rate (m):

  $$m = C_{d} A_{o} \left[ {\frac{{2\rho_{o} (\Delta p)_{o} }}{{1 - \beta^{4} }}} \right]^{1/2}$$
  $$\frac{\delta \beta }{\beta } = \left[ {\left( {\frac{{\delta d_{i} }}{{d_{i} }}} \right)^{2} + \left( {\frac{{\delta d_{o} }}{{d_{o} }}} \right)^{2} } \right]^{1/2} = \frac{\delta \beta }{\beta } = \left[ {\left( {\frac{0.1}{{81}}} \right)^{2} + \left( {\frac{0.1}{{40.5}}} \right)^{2} } \right]^{1/2} = 0.00276$$
  $$\frac{\delta m}{m} = \left[ {\left( {\frac{{\delta C_{d} }}{{C_{d} }}} \right)^{2} + \left( {\frac{{\delta A_{o} }}{{A_{o} }}} \right)^{2} + \left( {\frac{{\delta \rho_{o} }}{{2\rho_{o} }}} \right)^{2} + \left( {\frac{{\delta (\Delta P)_{o} }}{{2(\Delta P)_{o} }}} \right)^{2} + \frac{1}{4}\left( {\frac{4\delta \beta }{\beta }} \right)^{2} } \right]^{1/2}$$
  $$\frac{\delta m}{m} = \left[ {\left( {0.015} \right)^{2} + \left( {0.004938} \right)^{2} + \left( {\frac{0.0015559}{2}} \right)^{2} + \left( {\frac{0.01}{{2 \times 12.8}}} \right)^{2} + \frac{1}{4}\left( {4 \times 0.00276} \right)^{2} } \right]^{1/2} = \, 0.0{\text{1675 or 1}}.{675}\%$$

  where δC_d/C_d = 1.5% (from calibration chart of orifice meter).

• 7. Velocity of air in the duct (V):

  $${\text{V = }}\frac{{\text{m}}}{{{\uprho }_{{\text{a}}} {\text{WH}}}}$$
  $$\frac{\delta V}{V} = \left[ {\left( {\frac{\delta m}{m}} \right)^{2} + \left( {\frac{{\delta \rho_{a} }}{{\rho_{a} }}} \right)^{2} + \left( {\frac{\delta A}{A}} \right)^{2} } \right]^{1/2}$$
  $$\frac{\delta V}{V} = \left[ {\left( {0.016764} \right)^{2} + \left( {0.001559} \right)^{2} + \left( {0.004013} \right)^{2} } \right]^{1/2} = \, 0.0{\text{172 or 1}}.{72}\%$$

• 8. Useful heat gain (Q_u):

  $$Q_{u} = mC_{P} \left( {T_{o} - T_{i} } \right) = mC_{P} \left( {\Delta T} \right)$$
  $$\frac{{\delta Q_{u} }}{{Q_{u} }} = \left[ {\left( {\frac{\delta m}{m}} \right)^{2} + \left( {\frac{{\delta C_{P} }}{{C_{P} }}} \right)^{2} + \left( {\frac{{\delta \left( {\Delta T} \right)}}{\Delta T}} \right)^{2} } \right]^{1/2}$$

  The uncertainty in δC_p = 0.1, δμ = 0.001 × 10^–5, δk = 0.00001 and δPr = 0.001

  $$\Delta T = T_{o} - T_{i}$$
  $$\delta \left( {\Delta T} \right) = \left[ {\left( {\frac{\delta \Delta T}{{\delta T_{o} }}.\delta T_{o} } \right)^{2} + \left( {\frac{\delta \Delta T}{{\delta T_{i} }}.\delta T_{i} } \right)^{2} } \right]^{1/2} = \left[ {\left( {1 \times 0.25} \right)^{2} + \left( {1 \times 0.25} \right)^{2} } \right]^{1/2} = 0.353^{^\circ } {\text{C}}$$
  $$\left( {\frac{\delta \Delta T}{{\Delta T}}} \right) = \left( {\frac{0.353}{{4.96}}} \right)$$
  $$\frac{{\delta Q_{u} }}{{Q_{u} }} = \left[ {\left( {0.016764} \right)^{2} + \left( {\frac{0.1}{{1006.29}}} \right)^{2} + \left( {\frac{0.353}{{4.96}}} \right)^{2} } \right]^{1/2} = \, 0.0{\text{73116 or 7}}.{311}\%$$

• 9. Heat transfer coefficient (h):

  $$h = \frac{{Q_{u} }}{{A_{p} \left( {T_{P} - T_{m} } \right)}} = \frac{{Q_{u} }}{{A_{p} \Delta T_{1} }}$$
  $$\frac{\delta h}{h} = \left[ {\left( {\frac{{\delta Q_{u} }}{{Q_{u} }}} \right)^{2} + \left( {\frac{{\delta A_{p} }}{{A_{p} }}} \right)^{2} + \left( {\frac{{\delta \left( {\Delta T_{1} } \right)}}{{\Delta T_{1} }}} \right)^{2} } \right]^{1/2} = \left[ {\left( {0.073116} \right)^{2} + \left( {0.001054} \right)^{2} + \left( {\frac{0.353}{{14.4}}} \right)^{2} } \right]^{1/2} = \, 0.0{\text{7711 or 7}}.{711}\%$$

• 10. Nusselt number (Nu):

  $$Nu = \frac{{hD_{h} }}{k}$$
  $$\frac{\delta Nu}{{Nu}} = \left[ {\left( {\frac{\delta h}{h}} \right)^{2} + \left( {\frac{{\delta D_{h} }}{{D_{h} }}} \right)^{2} + \left( {\frac{\delta k}{k}} \right)^{2} } \right]^{1/2} = \left[ {\left( {0.07712} \right)^{2} + \left( {0.00369} \right)^{2} + \left( {\frac{0.00001}{{0.026109}}} \right)^{2} } \right]^{1/2} = \, 0.0772 \, or \, 7.72\%$$

• 11. Reynolds number (Re):

  $${\text{Re}} = \frac{{\rho_{a} VD_{h} }}{\mu }$$
  $$\frac{{\delta {\text{Re}} }}{{\text{Re}}} = \left[ {\left( {\frac{{\delta \rho_{a} }}{{\rho_{a} }}} \right)^{2} + \left( {\frac{\delta V}{V}} \right)^{2} + \left( {\frac{{\delta D_{h} }}{{D_{h} }}} \right)^{2} + \left( {\frac{\delta \mu }{\mu }} \right)^{2} } \right]^{1/2} = \left[ {\left( {0.001559} \right)^{2} + \left( {0.01732} \right)^{2} + \left( {0.00369} \right)^{2} + \left( {\frac{{0.001 \times 10^{ - 5} }}{{1.8346 \times 10^{ - 5} }}} \right)^{2} } \right]^{1/2} = \, 0.01776 \, or \, 1.77\%$$

• 12. Friction factor (f):

  $$f = \frac{{D_{h} \left( {\Delta P_{d} } \right)}}{{2L_{f} V^{2} \rho }}$$
  $$\begin{aligned} \frac{\delta f}{f} & = \left[ {\left( {\frac{{\delta D_{h} }}{{D_{h} }}} \right)^{2} + \left( {\frac{{\delta \left( {\Delta P} \right)_{d} }}{{\left( {\Delta P} \right)_{d} }}} \right)^{2} + \left( {\frac{\delta L}{L}} \right)^{2} + \left( {\frac{2\delta V}{V}} \right)^{2} + \left( {\frac{{\delta \rho_{a} }}{{\rho_{a} }}} \right)^{2} } \right]^{1/2} \\ & = \left[ {\left( {0.00369} \right)^{2} + \left( {\frac{0.1}{{28.8}}} \right)^{2} + \left( \frac{1}{1000} \right)^{2} + \left( {2 \times 0.01732} \right)^{2} + \left( {0.001559} \right)^{2} } \right]^{1/2} \\ & = 0.035 \, or \, 3.50\% \\ \end{aligned}$$