Answer:
(a) h₁ = 204.45 W/m²k
(b) h₀ = 46.80 W/m².k
(c) T = T = 15.50°C
Explanation:
Given Data;
Diameter = 12mm
Length = 25 m
Entry temperature = 200°C
Flow rate = 0.006 kg/s
velocity = 2.5 m/s.
Step 1: Calculating the mean temperature;
(200 + 15)/2
Mean temperature = 107.5°C = 380.5 K
The properties of air at mean temperature 380.5 K are given as:
v = 24.2689*10⁻⁶m²/s
a = 35.024*10⁻⁶m²/s
μ = 221.6 *10⁻⁷Ns/m²
k = 0.0323 W/m.k
Cp = 1012 J/kg.k
Step 2: Calculating the prantl number using the formula;
Pr = v/a
= 24.2689*10⁻⁶/ 35.024*10⁻⁶
= 0.693
Step3: Calculating the reynolds number using the formula;
Re = 4m/πDμ
= 4 *0.006/π*12*10⁻³ * 221.6 *10⁻⁷
= 0.024/8.355*10⁻⁷
= 28725
Since Re is greater than 2000, the flow is turbulent. Nu becomes;
Nu = 0.023Re^0.8 *Pr^0.3
Nu = 0.023 * 28725^0.8 * 0.693^0.3
= 75.955
(a) calculating the heat transfer coefficient:
Nu = hD/k
h = Nu *k/D
= (75.955 * 0.0323)/12*10^-3
h = 204.45 W/m²k
(b)
Properties of air at 15°C
v = 14.82 *10⁻⁶m²/s
k = 0.0253 W/m.k
a = 20.873 *10⁻⁶m²/s
Pr(outside) = v/a
= 14.82 *10⁻⁶/20.873 *10⁻⁶
= 0.71
Re(outside) = VD/v
= 2.5 * 12*10⁻³/14.82*10⁻⁶
=2024.29
Using Zakauskus correlation,
Nu = 0.26Re^0.6 * Pr^0.37 * (Pr(outside)/Pr)^1/4
= 0.26 * 2024.29^0.6 * 0.71^0.37 * (0.71/0.693)^1/4
= 22.199
Nu = h₀D/k
h₀ = Nu*k/D
= 22.199* 0.0253/12*10⁻³
h₀ = 46.80 W/m².k
(c)
Calculating the overall heat transfer coefficient using the formula;
1/U =1/h₁ +1/h₀
1/U = 1/204.45 + 1/46.80
1/U = 0.026259
U = 1/0.026259
U = 38.08
Calculating the temperature of the exhaust using the formula;
T -T₀/T₁-T₀ = e^-[uπDL/Cpm]
T - 15/200-15 = e^-[38.08*π*12*10⁻³*25/1012*0.006]
T - 15/185 = e^-5.911
T -15 = 185 * 0.002709
T = 15+0.50
T = 15.50°C
