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Hot engine oil with a heat capacity rate of 4440 W/K (product of mass flow rate and specific heat) and an inlet temperature of 150ºC flows through a double pipe heat exchanger.

The double pipe heat exchanger is constructed using a 1.5-m-long copper pipe (k = 250 W/m·K) with an inner tube of inside diameter 2 cm and outside tube diameter of 2.25 cm. The inner diameter of the outer tube of the double pipe heat exchanger is 6 cm.

Oil flowing at a rate of 2 kg/s through the inner tube exits the heat exchanger at a temperature of 50ºC. The cold fluid, i.e., water, enters the heat exchanger at 20ºC and exits at 108ºC. Assume the fouling factor on the oil side and water side to be 0.00015 m²·K/W and 0.0001 m²·K/W, respectively.

Determine the overall heat transfer coefficient on the inner and outer surface of the copper tube.

Answer :

The overall heat transfer coefficient on the inner and outer surface of the copper tube is 240.49 W/m2.K.

How did we get the value?

To determine the overall heat transfer coefficient on the inner and outer surface of the copper tube, we need to calculate the individual heat transfer coefficients and then find the overall coefficient using the following formula:

1 / U = 1/ h i + θ i / k + θ o / k + 1 / h o

Where h i is the heat transfer coefficient on the inner surface, h o is the heat transfer coefficient on the outer surface, θ i is the f o u l i n g factor on the inner surface, θ o is the f o u l i n g factor on the outer surface, and k is the thermal conductivity of the copper tube.

We can calculate the mass flow rate of the oil using the following formula:

m = p * V * A

Where p is the density of the oil, V is the velocity of the oil, and A is the cross-sectional area of the inner tube.

The density of the oil can be calculated using the following formula:

p = m / V

Where m is the mass flow rate of the oil, and V is the volume flow rate of the oil.

The volume flow rate of the oil can be calculated using the following formula:

V = Q / m C p

Where Q is the heat flow rate, Cp is the specific heat capacity of the oil, and m is the mass flow rate of the oil.

We can calculate the heat flow rate using the following formula:

Q = m C p (T h i - T h o)

Where T h i is the in let temperature of the oil, and T h o is the outlet temperature of the oil.

Using these formulas and the given values, we can calculate the following:

The mass flow rate of the oil:

m = p * V * A = (850 kg/m3) * (2 m/s) * (π * (0.02 m)2 / 4) = 0.021 m3/s

The heat flow rate:

Q = m C p (Thi - Tho) = (2 kg/s) * (4440 W/K) * (150°C - 50°C) = 532.8 kW

The volume flow rate of the oil:

V = Q/m C p = (532.8 kW) / ((2 kg/s) * (4440 W/K)) = 0.06 m3/s

The R e y n o l d s number of the oil:

Re = p * V * D / μ

Where D is the diameter of the inner tube, and μ is the vi s c o s i t y of the oil.

μ = 0.035 Ns/m2 (v i s c o s i t y of engine oil at 150°C)

Re = (850 kg/m3) * (2 m/s) * (0.02 m) / (0.035 Ns/m2) = 968.57

The N u s s e l t number for the oil:

Nu = 0.023 * Re 0.8 * Pr0.4

Where Pr is the P r a n d t l number of the oil, which can be assumed to be 50 at the average temperature of the oil.

N u = 0.023 * (968.57)0.8 * (50)0.4 = 192.75

The heat transfer coefficient on the inner surface of the copper tube:

hi = N u * k / D = (192.75) * (250 W/m.K) / (0.02 m) = 240937.5 W /m2.K

The f o u l ing factor on the inner surface of the copper tube:

θ i = 0.00015 m2.K/W

The thermal resistance on the inner surface of the copper tube:

Ri = θ i / (π * D * L) = (0.00015 m2.K/W) / (π * 0.02 m * 1.5 m) = 0.00265 K/W

The thermal resistance on the outer surface of the copper tube:

Ro = θ o / (π * D * L) = (0.0001 m2.K/W) / (π * 0.06 m * 1.5 m) = 0.00145 K/W

The heat transfer coefficient on the outer surface of the copper tube:

h o = k / (D * R o)

h o = (250 W /m.K) / (0.06 m * 0.00145 K/W) = 287.36 W /m2.K

Now we can calculate the overall heat transfer coefficient:

1/U = 1 / h i + θ i / k + θ o / k + 1 / h o

1/U = (1/240937.5) + (0.00015/250) + (0.0001/250) + (1/287.36)

1/U = 0.000004154

U = 240.49 W/ m2.K

Therefore, the overall heat transfer coefficient on the inner and outer surface of the copper tube is 240.49 W/ m2.K.

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