Transient Heat Transfer Analysis in Insulated Pipe with Constant and Time-Dependent Heat Flux for Solar Convective Furnace

Abstract

The paper deals with heat transfer during the transportation of the hot air from receiver via an insulated pipe to solar convective furnace for metal processing. In the developed concept of solar convective furnace (SCF) hot air at a temperature of 300–600 °C is required for a duration of about 3–6 h. However, the availability of solar irradiance is 8–10 h per day, whereas, the typical required time for achieving the steady condition is in the order of hours. Thus, prediction of time-dependent air temperature development at the outlet of insulated pipe is required for a given flow condition. Considering these aspects, the developed and validated transient heat transfer analysis tool is used to analyze effect of (a) pipe length and thickness of insulation, (b) mass-flow-rate of air, (c) constant and variable inlet air temperature and (d) preheating of pipe. Following are the broad observations based on the performed analysis: (i) time to reach steady state for the considered insulated pipe and the ratio of the heat loss to input reduces with increasing mass-flow-rate; (ii) increasing insulation thickness beyond critical thickness reduces the heat loss that results in a higher temperature at outlet. However, it increases the required time to reach the steady state condition; (iii) the achieved maximum temperature corresponding to a solar irradiance is forward shifted in time.

Appendix

Equations (20), (21) and (22) represents the discretized form of the equations in fluid, solid and insulation domains and Eqs. (23) and (24) represents the discretized equations in solid and insulation domain respectively.

$$T_{{{\text{m}}_{i} }}^{n + 1} = T_{{{\text{m}}_{i} }}^{n} (1 - \left( {\left( {b + 1} \right)/c} \right) + T_{{{\text{s}}_{i} }}^{n} \left( {b/c} \right) + T_{{{\text{m}}_{i} }}^{n - 1} /\left( c \right)$$

(20)

$$T_{{{\text{s}}_{i} }}^{n + 1} = T_{{{\text{s}}_{i} }}^{n} *C\left( 3 \right) + T_{{{\text{s}}_{i + 1} }}^{n} *\tau_{\text{s}} + T_{{{\text{s}}_{i - 1} }}^{n} *\tau_{\text{s}} + T_{{{\text{m}}_{i} }}^{n} *C\left( 1 \right) + T_{{{\text{i}}_{i,j = 2} }}^{n} *C\left( 2 \right)$$

(21)

$$T_{ii,j}^{n + 1} = T_{ii,j}^{n} *g\left( 2 \right) + T_{ii,j - 1}^{n} *\tau_{i} + T_{ii,j + 1}^{n} *g\left( 3 \right)$$

(22)

$$T_{{{\text{s}}_{i} }}^{n + 1} = T_{{{\text{s}}_{i} }}^{n} *D\left( 4 \right) + T_{{{\text{s}}_{i + 1} }}^{n} *\tau_{\text{s}} + T_{{{\text{s}}_{i - 1} }}^{n} *\tau_{\text{s}} + T_{{{\text{m}}_{i} }}^{n} *D\left( 1 \right) + T_{{{\text{i}}_{{_{i} ,j = 2}} }}^{n} *D\left( 2 \right) + D\left( 3 \right) *T_{\text{a}}$$

(23)

$$T_{{{\text{i}}_{{i,{\text{Nz}}}} }}^{n + 1} = T_{{{\text{i}}_{{i,{\text{Nz}}}} }}^{n} *e\left( 3 \right) + T_{{{\text{i}}_{{i,{\text{Nz}} - 1}} }}^{n} *2\tau_{\text{i}} + T_{\text{a}} *e\left( 2 \right)$$

(24)

where

$$\begin{aligned} b & = \left( {\frac{{2\pi r_{\text{s}} h_{\text{f}} \Delta z}}{{m_{\text{f}} c_{\text{pf}} }}} \right),{\text{c}} = \left( {\frac{{\uprho_{\text{f}} \pi r_{\text{s}}^{2} \Delta {\text{z}}}}{{m_{\text{f}} \Delta t}}} \right), \alpha_{\text{s}} = \frac{{\rho_{\text{s}} C_{\text{ps}} }}{{K_{\text{s}} }},\tau_{\text{s}} = \frac{{\left( {\Delta z} \right)^{2} }}{{\alpha_{\text{s}} \Delta t}},C\left( 1 \right) \\ & = \frac{{h_{\text{f}} \tau_{\text{s}} \left( {\Delta z} \right)^{2} }}{{K_{\text{s}} \delta_{\text{s}} }} ,C\left( 2 \right) = \frac{{K_{\text{i}} \tau_{\text{s}} \left( {\Delta z} \right)^{2} }}{{K_{\text{s}} \delta_{\text{s}} \left( {\Delta r} \right)}},e\left( 1 \right) = \frac{{2\left( {\Delta r} \right)^{2} \tau_{\text{i}} }}{{K_{\text{i}} r_{\text{i}} }},C\left( 3 \right) \\ & = 1 - 2\tau_{\text{s}} - C\left( 1 \right) - C\left( 2 \right), D\left( 1 \right) = \frac{{h_{\text{f}} \tau_{\text{s}} \left( {\Delta z} \right)^{2} }}{{K_{\text{s}} \delta_{\text{s}} }}, D\left( 2 \right) \\ & = \frac{{K_{\text{i}} \tau_{\text{s}} \left( {\Delta z} \right)^{2} }}{{K_{\text{s}} \delta_{\text{s}} \left( {\Delta r} \right)}}, e\left( 2 \right) = h_{\text{ex}} *e\left( 1 \right),D\left( 3 \right) = \frac{{2\tau_{\text{s}} h_{\text{ex}} \left( {\Delta z} \right)}}{{K_{\text{s}} \delta_{\text{s}} }}, D\left( 4 \right) \\ & = 1 - 2\tau_{\text{s}} - D\left( 1 \right) - D\left( 2 \right) - D\left( 3 \right), \tau_{\text{i}} = \frac{{\left( {\Delta r} \right)^{2} }}{{\alpha_{\text{i}} \Delta t}}, e\left( 3 \right) \\ & = 1 - 2\tau_{\text{i}} \frac{\Delta r}{{r_{\text{i}} }} - \frac{{2\Delta r\tau_{\text{i}} h_{\text{ex}} }}{{K_{\text{i}} }},g\left( 1 \right) = \frac{\Delta r}{r}, g\left( 2 \right) \\ & = 1 - 2\tau_{\text{i}} - g\left( 1 \right) *\tau_{\text{i}} ,g\left( 3 \right) = \left( {1 + g\left( 1 \right)} \right) *\tau_{\text{i}} \\ \end{aligned}$$