Q3: An air of an average velocity um 1m/s flows in in a fully developed region...

  

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Q3: An air of an average velocity um 1m/s flows in in a fully developed region of a circular tube filled with metal foam under the second boundary condition with a uniform flux qw = 500W/m². The radius of the tube ro=10cm. Pore diameter d and porosity & of porous media is 0.5cm and 0.5, respectively. The fluid density pf = 1.205kg/ m³, thermal conductivity k, = 0.0259W/(mK), kinetic viscosity = 1.81 x 10-5, heat capacity C 1005J/(kg K) and thermal conductivity of solid phase k, = 398W/(mK). The microstructure isn't considered here and the momentum equation can be modified with Brinkman-Darcy model. = d²e³ where the permeability K = 150(1-8)² Rc = The effective thermal conductivity of solid phase can be determined by: kse where R₁ = 42 e-22 [2e²+(1-e)]k, +{4-[2e²+ m2(1-e)]}k, e²k +(2-e²)k, dp =-- -um=- dz K (√2-2e)² 272²k. (1-2√2e)+2(√2-2e-n2² (1-2√2e)) k e=0.399, λ = K 5 √22-28-√2e³ 7(3-e-4e√2) R₂ - RD = 2e e²k +(4-e²) k √2 2(RA+RB+RC+RD) 1) Solve the temperature distribution and velocity distribution. 2) If the porosity changes with R linearly: & = 0.3 +0.01R, where R is the distance to the center of the tube, solve the temperature distribution and velocity distribution. 3) Considering the thermal performance of a counter-flow tube-in-tube heat exchanger as shown in Fig. 1. The heat exchanger consists of two concentric pipes forming an inner section and an outer annular section, both filled with metal foams, which have the porosity of 0.5 and 0.6, respectively. The radius of the outer pipe is 20 cm. Fluid flows axially through both sections in a counter-flow arrangement. The outer pipe is assumed to be perfectly insulated so there is no heat transfer between the pipe's outer surface area and the surroundings. The heat flux through the wall of the inner pipe is assumed to be constant so the thermal conditions of the inner surface of the outer section can be assumed to constant heat flux. Solve the temperature distribution and velocity distribution. 4) Considering the heat performance of the finned heat exchanger as shown in Fig. 2, the inner section is filled with metal foam and outer section is filled with 20 longitudinal copper fins. The depth of the fins is 2.5 mm and fin thickness is 0.075 mm. Solve the temperature distribution and compare it with question 3). 5) Optimize the structure (like the configuration, tube size, pore size, porosity and so on) based on the overall performance in terms of heat transfer and flow resistance. 0.2m 0.4m Fig. 1 0.2m 0.4m Fig. 2 Q3: An air of an average velocity um 1m/s flows in in a fully developed region of a circular tube filled with metal foam under the second boundary condition with a uniform flux qw = 500W/m². The radius of the tube ro=10cm. Pore diameter d and porosity & of porous media is 0.5cm and 0.5, respectively. The fluid density pf = 1.205kg/ m³, thermal conductivity k, = 0.0259W/(mK), kinetic viscosity = 1.81 x 10-5, heat capacity C 1005J/(kg K) and thermal conductivity of solid phase k, = 398W/(mK). The microstructure isn't considered here and the momentum equation can be modified with Brinkman-Darcy model. = d²e³ where the permeability K = 150(1-8)² Rc = The effective thermal conductivity of solid phase can be determined by: kse where R₁ = 42 e-22 [2e²+(1-e)]k, +{4-[2e²+ m2(1-e)]}k, e²k +(2-e²)k, dp =-- -um=- dz K (√2-2e)² 272²k. (1-2√2e)+2(√2-2e-n2² (1-2√2e)) k e=0.399, λ = K 5 √22-28-√2e³ 7(3-e-4e√2) R₂ - RD = 2e e²k +(4-e²) k √2 2(RA+RB+RC+RD) 1) Solve the temperature distribution and velocity distribution. 2) If the porosity changes with R linearly: & = 0.3 +0.01R, where R is the distance to the center of the tube, solve the temperature distribution and velocity distribution. 3) Considering the thermal performance of a counter-flow tube-in-tube heat exchanger as shown in Fig. 1. The heat exchanger consists of two concentric pipes forming an inner section and an outer annular section, both filled with metal foams, which have the porosity of 0.5 and 0.6, respectively. The radius of the outer pipe is 20 cm. Fluid flows axially through both sections in a counter-flow arrangement. The outer pipe is assumed to be perfectly insulated so there is no heat transfer between the pipe's outer surface area and the surroundings. The heat flux through the wall of the inner pipe is assumed to be constant so the thermal conditions of the inner surface of the outer section can be assumed to constant heat flux. Solve the temperature distribution and velocity distribution. 4) Considering the heat performance of the finned heat exchanger as shown in Fig. 2, the inner section is filled with metal foam and outer section is filled with 20 longitudinal copper fins. The depth of the fins is 2.5 mm and fin thickness is 0.075 mm. Solve the temperature distribution and compare it with question 3). 5) Optimize the structure (like the configuration, tube size, pore size, porosity and so on) based on the overall performance in terms of heat transfer and flow resistance. 0.2m 0.4m Fig. 1 0.2m 0.4m Fig. 2

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