Air enters a horizontal, constant-diameter heating duct operating at steady state at 290 K, 1 bar, with a volumetric flow rate of 0.25 m3/s, and exits at 325 K, 0.95 bar. The flow area is 0.04 m2. Assuming the ideal gas model with k 5 1.4 for the air, determine:

(a) the mass flow rate, in kg/s,

(b) the velocity at the inlet and exit, each in m/s, and

(c) the rate of heat transfer, in kW.

A) Mass flow rate = 0.3004 Kg/s

B) Velocity at Inlet = 6.25 m/s

Velocity at exit = 7.3725 m/s

C) Rate of heat transfer = 12.858 Kw

Explanation:

T1 = 290K; P1 = 1 bar = 100 KPa

T2 = 325K; P2 = 0.95 bar = 95 KPa

A = 0.04 m²; k = 1.4

Molar mass of air = 28.97 Kg/Kmol

R = 8.314 J/molK

A) Since we are dealing with a steady state mass flow rate through an open system, we will treat this as an ideal gas. Thus;

PV = nRT

n=m/M where m = 1

Thus; V1 = RT1/MP1 = (8.314 X 290) / (28.97 x 100) = 0.8323 m³/Kg

Now, mass flow rate is given by;

Mass flow rate (m') = Volumetric flow rate (V') / Volume (v)

Thus; Mass flow rate (m') = 0.25/0.8323 = 0.3004 Kg/s

From Gay lussacs law; P1V1/T1 = P2V2/T2

So to find the volume at the exit which is V2, let's make V2 the subject of the formula;

(P1V1T2) / (P2T1) = V2

So; V2 = (100 x 0.8323 x 325) / (95 x 290) = 0.9818 m³/kg

B) we know that velocity = volumetric flow rate/area

Thus;

At inlet; Velocity (Vi) = 0.25/0.04 = 6.25m/s

At exit; Velocity (Ve) = Volumetric flow rate at exit / Area.

We don't know the volumetric flow at exit so let's look for it. from earlier, we saw that;

volumetric flow rate/Volume = mass flow rate

And rearranging, volumetric flow rate (V') = mass flow rate (m) x volume (v)

So V' = 0.3004 x 0.9818 = 0.2949 Kg/s

So, Ve = 0.2949/0.04 = 7.3725 m/s

C) The steady state equation when potential energy is neglected is given by the formula;

Q' = m'Cp (T2 - T1) + (m'/2) { (Ve) ² - (Vi) ²}

Where Q' is the rate of heat transfer.

Cp is unknown. The formula to find Cp is given as;

Cp = KR / (K-1)

Since we are dealing with change in enthalpy here, the gas constant R will be expressed per molecular mass of the air and so R = 0.287 KJ/kg. k

Cp = (1.4 x 0.287) / (1.4 - 1) = 1.0045 Kj/KgK

And so, Q' = (0.3004 x 1.0045) (325 - 290) + (0.3004/2) { (7.3725) ² - (6.25) ²} = 10.5613 + 2.2967 = 12.858 Kw

a) m = 0.3003 kg/s

b) Vel1 = 6.25 m/s, Vel2 = 7.3725 m/s

c) 12.845 KW

Explanation:

a)

using ideal gas law:

PV = nRT

since, n = no. of moles = m/M

therefore,

PV = (m/M) RT

P1 v1 = RT1/M

where,

P1 = inlet pressure = 1 bar = 100000 Pa

v1 = specific volume at inlet = ?

R = universal gas constant = 8.314 KJ/Kmol. k

T1 = inlet temperature = 290 K

M = Molecular mass of air = 28.9628 Kg/kmol

Therefore,

v1 = (8.314 KJ/Kmol. k) (290 k) / (28.9628 kg/kmol) (100 KPa)

v1 = 0.83246 m³/kg

Now, the mass flow rate can be given as:

Mass Flow Rate = (Volume Flow Rate) / (v1)

Mass Flow Rate = (0.25 m³/s) / (0.83246 m³/kg)

Mass Flow Rate = 0.3003 kg/s

b)

using general gas equation to find the specific volume at the exit first, we get:

P1v1/T1 = P2v2/T2

v2 = P1 v1 T2/T1 P2

where,

P1 = inlet pressure = 1 bar

P2 = exit pressure = 0.95 bar

T1 = inlet temperature = 290 k

T2 = exit temperature = 325 k

v1 = specific volume at inlet = 0.83246 m³/kg

v2 = specific volume at exit = ?

Therefore,

v2 = (1 bar) (0.83246 m³/kg) (325 k) / (290 k) (0.95 bar)

v2 = 0.98203 m³/kg

Now, for velocity, we use formula:

Vel = v/A

where,

A = Area = 0.04 m²

For inlet:

Vel1 = Inlet Volume flow Rate/A = (0.25 m³/s) / (0.04 m²)

Vel1 = 6.25 m/s

For exit:

Vel2 = Exit Volume flow Rate/A = (Mass Flow Rate) (v2) / (0.04 m²)

Vel2 = (0.98203 m³/kg) (0.3003 kg/s) / 0.04 m²

Vel2 = 7.3725 m/s

c)

using first law of thermodynamics, with no work done, we can derive the formula given below:

Q = m (h2 - h1) + (m/2) (Vel2² - Vel1²)

where,

Q = rate of heat transfer

m = mass flow rate = 0.3003 kg/s

m (h2 - h1) = change in enthalpy = mCpΔT, for ideal gas

Vel1 = 6.25 m/s

Vel2 = 7.3725 m/s

Also,

Cp in this case will be function of temperature and given as:

Cp = KR / (K-1)

where,

K = 1.4

R = Gas constant per molecular mass of air = 0.28699 KJ/kg. k

Therefore,

Cp = 1.0045 KJ/kg. k

Now, using the values in the expression of first law of thermodynamics, we get:

Q = (0.3003 kg/s) (1.0045 KJ/kg. k) (35 K) + [ (0.3003 kg/s) / 2][ (7.3725 m/s) ² - (6.25 m/s) ²]

Q = 10.55 KW + 2.29 KW

Q = 12.845 KW