An 8-m long, uninsulated square duct of cross section 0.2m x 0.2m and relative roughness 10^-3 passes through the attic space of a house.. Hot air (80°C) enters an 8 m long un-insulated square duct (cross section 0.2 m x 0.2 m) that passes through the attic of a house at a rate of 0.15 m^3 /s. The duct is isothermal at a temperature of 60°C. Determine the rate of heat loss from the duct to the attic space and the pressure difference between the inlet and outlet sections of the duct.

the rate of heat loss from the duct to the attic space = 1315.44 W

the pressure difference between the inlet and outlet sections of the duct  = 7.0045 N/m²

Explanation:

We know that properties of air 80⁰C  and 1atm  (from appendix table) are;

density p = 0.9994 kg/m³, Specifice heat Cp = 1008 J/kg.⁰C

Thermal conductivity k = 0.02953 W/m.⁰C, Prandtl number Pr = 0.7154,

Kinematic viscosity v = 2.097 × 10⁻⁵ m²/s

haven gotten that, we calculate the hydraulic diameter of square duct

Dh = 4Ac / P      { Ac = is cross sectional area of duct and P = perimeter}

now we substitute a² for Ac and 4a for P ( we know from the question that a = 0.2 m)

Dh = 4a² / 4a

Dh = 4(0.2)² / 4(0.2)

Dh = 0.2 m

Now we calculate the average velocity of air

Vₐ = Vˣ / Ac        { vˣ = volume flow rate of air}

Vₐ = Vˣ / a²      { Ac = a² }, we know that a = 0.2m₂, Vˣ = 0.15 m³

Vₐ = 0.15 / (0.2)²

Vₐ = 3.75 m/s

Next we calculate the Reynolds number

Re = Vₐ Dh / V

Re = (3.75 × 0.2) / 2.097× 10⁻⁵

Re = 35765.379

The  Reynolds number IS GREATER than 10,000

so the flow is turbulent and entry length in this case is nearly 10 times the hydraulic diameter

Lh ≈ Lt ≈ 10D

= 10 × 0.2

= 2m

As this length is quite small when compared to the total of tube, we assume fully developed flow for the entire tube length.

Now we calculate the Nusselt number from this relation;

Nu =  0.023 Re⁰'⁸ Pr⁰'³

so we substitute for Re and Pr

Nu = 0.023(35765.379)⁰'⁸ (0.7154)⁰'³

Nu = 91.4

Now calculate the convective heat transfer coefficient

h = Nu × K/ Dh

we substitute

h = 91.4 × 0.02953 W/m.°C / 0.2 m

h = 13.5 W/m².°C

We calculate the surface area of the square duct

Aₓ = 4aL       { L= length of duct}

we substitute

Aₓ = 4 × 0.2 × 8

Aₓ = 6.4 m²

Mass flow rate of air

m = pVˣ

we substitute again ( from our initials)

m = 0.9994 kg/m₃ × 0.15 m³/s

m= 0.150 kg/s

We calculate the exit temperature of the air from the duct

Te = Ts - (Ts -Ti) exp ( - hAₓ / mCp)

we know that

Ts = 60°C , Ti = 80°C, h = 13.5 W/m².°C , Aₓ = 6.4m², m = 0.150 kg/s , Cp = 1008 J/kg.°C

we substitute

Te = 60 - (60-80) exp(- ((13.5 × 6.4)/(0.15 × 1008))

Te = 71.3°

Now we calculate the rate of heat loss from the duct.

Q = mCp ( Ti -Te )

we substitute again

Q = 0.150 × 1008 × ( 80 - 71.3 )

Q = 1315.44 W

Next we calculate the estimated friction factors by using Haaland equation

1/√f = - 1.8log₁₀ [ 6.9/Re + (E/D)/3.7)¹'¹¹]

we know that E/D = relative roughness = 10⁻³

we substitute

so

1/√f = - 1.8log₁₀ [ (6.9/35765.379) + ( 10⁻³/3.7)¹'¹¹]

1/√f = - 1.8log₁₀ { 0.000192924 + 0.00010947}

1/√f = - 1.8log₁₀ 0.000302324  

√f =   1/6.334

f = (1/6.334)²

f = 0.02492

We calculate the pressure difference between inlet and outlet sections of the duct

ΔPl = fLPVa² / Dh × 2

ΔPl = {0.02492 × 8 × 0.9994 × (3.75)²} / 0.2 × 2

ΔPl = 2.8018 / 0.4

ΔPl = 7.0045 N/m²

Therefore pressure deference is 7.0045 N/m²

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