**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**