Step-by-step explanation:
To determine the heat transfer rate of the system and the length of the tube required for fully developed flow, we can use the heat transfer rate equation and the concept of fully developed flow.
The heat transfer rate (Q) can be calculated using the equation:
Q = m_dot * Cp * (T_out - T_in)
where:
Q is the heat transfer rate (W)
m_dot is the mass flow rate of water (kg/s)
Cp is the specific heat capacity of water (J/kg·K)
T_out is the outlet temperature of water (°C)
T_in is the inlet temperature of water (°C)
The mass flow rate (m_dot) can be calculated using the volumetric flow rate (Q_dot) and the density of water (rho):
m_dot = Q_dot * rho
The volumetric flow rate (Q_dot) is given as 0.05 m³/s.
The density of water (rho) can be approximated as 1000 kg/m³.
The specific heat capacity of water (Cp) is approximately 4186 J/kg·K.
Given:
Inlet temperature (T_in) = 25 °C
Outlet temperature (T_out) = 65 °C
Now, let's calculate the heat transfer rate (Q) using the given values:
m_dot = Q_dot * rho
m_dot = 0.05 m³/s * 1000 kg/m³
m_dot = 50 kg/s
Q = m_dot * Cp * (T_out - T_in)
Q = 50 kg/s * 4186 J/kg·K * (65 °C - 25 °C)
Once you calculate Q, you will have the heat transfer rate for the system.
To determine the length of the tube required for fully developed flow, we need to consider the concept of fully developed flow. In fully developed flow, the velocity profile across the tube diameter becomes fully developed, and the flow properties remain constant along the tube's length.
For a smooth tube, the length required for fully developed flow can be approximated using the following formula:
L = (7.6 * D) * (Re * Pr)^0.4
where:
L is the length required for fully developed flow (m)
D is the tube diameter (m)
Re is the Reynolds number
Pr is the Prandtl number
The Reynolds number (Re) and Prandtl number (Pr) can be calculated as follows:
Re = (rho * V * D) / mu
Pr = Cp * mu / k
where:
rho is the density of water (kg/m³)
V is the average velocity of water (m/s)
mu is the dynamic viscosity of water (Pa·s)
k is the thermal conductivity of water (W/m·K)
Please note that the above calculations assume fully developed flow and neglect any pressure losses or other effects. For more accurate results, detailed calculations and considerations may be necessary.