Application of First Law in a PC Fan

The Control Volume

The whole PC case is considered the control volume. Air enters via the intake fans, combines with the internal air, and exits via the exhaust fans.

Assumption

The key assumption is that first there is steady state flow. The mass flow level and other properties at both inlet and outlet are constant over time. Another assumption is that there is no heat generated within the control volume. Only heat movement between the air and the internal element occurs. The third assumption is there are uniform inlet and outlet conditions. The air velocity and temperature are uniform across both inlet and outlet areas. Lastly, there is the assumption that there is no heat loss to the surroundings. The air moves away from every heat generated by a motherboard and fan.

Local Conditions:

The local average temperature will use the typical room temperature of (20^oc), while the atmospheric pressure is (101,325 kPa) and therefore they form (T₁) and (P₁) respectively.

Minimum Required Fan Inlet Area

To calculate the minimum required fan inlet area, it is important to ensure the incoming air velocity constraint is achieved

Continuity equation is

Mass flow rate of air (m)= Density of air (p) x inlet air velocity (V_in) x inlet area (A_in)

M= (P x V_in x A_in)

Energy Balance- The first law of thermodynamics for an open system is

Q-W= m. (h_out– h_in)

Ideal Gas Law is

P= P/ R.T

For the air :

P is 101325 Pa

R = 287 J/kg

T_in= 20^o C= 293k

Therefore, the density of air (p) is calculated as

P= (101325 / 287.293)

P= 1.204 kg/m³

For simplicity, applying cp for air is

H= (Cp x T)

C_p= 1005 J/Kg

Therefore

98= m. cp. (T_out – T_in)

Calculating for M

98= M x 1005. (305-293)

98 = m * 1005 (12)

M= 98/ 1005.12

= 0.0081 kg/ s

Minimum Required Fan Inlet Area:

Using the equation

0.0081= 1.204 x 1.5. A_in

A_in = (0.0081/ 1.204 x 1.5)

= 0.0045 m²

Converting Meter to centimetres

A_in = (0.0045 x 10⁴)

= 45 cm ²