Answer:
To solve this problem, we need to use the Reynolds number, the Prandtl number, and the Darcy friction factor. The Reynolds number is a dimensionless number that describes the flow regime of the fluid, the Prandtl number characterizes the ratio of momentum diffusivity to thermal diffusivity, and the Darcy friction factor describes the frictional losses in the pipe.
(a)
For pressurized water at 366 K and a flow rate of 0.01 kg/s, we can calculate the mean velocity as follows:
v = m_dot / (rho * A), where
m_dot = 0.01 kg/s (mass flow rate)
rho = 997 kg/m^3 (density of water at 366 K)
A = pi * (0.02 m)^2 / 4 = 0.000314 m^2 (cross-sectional area of the tube)
v = 0.01 / (997 * 0.000314) = 3.18 m/s
The Reynolds number for water is:
Re = (rho * v * D) / mu, where
D = 0.02 m (diameter of the tube)
mu = 0.000283 Pa s (dynamic viscosity of water at 366 K)
Re = (997 * 3.18 * 0.02) / 0.000283 = 2.24e5
Using the Reynolds number, we can estimate the hydrodynamic entry length using the following equation:
L_hyd = 0.05 * Re * D
L_hyd = 0.05 * 2.24e5 * 0.02 = 224 m
For the thermal entry length, we use the following equation:
L_th = 0.05 * Re * Pr * D / Nu
where Pr is the Prandtl number and Nu is the Nusselt number. For water at 366 K, Pr = 5.5 and we can estimate Nu using the Dittus-Boelter correlation:
Nu = 0.023 * Re^(4/5) * Pr^(0.4)
Nu = 0.023 * (2.24e5)^(4/5) * 5.5^(0.4) = 244
L_th = 0.05 * 2.24e5 * 5.5 * 0.02 / 244 = 3.6 m
For engine oil (unused) at 366 K and a flow rate of 0.01 kg/s, we can calculate the mean velocity as follows:
v = m_dot / (rho * A), where
m_dot = 0.01 kg/s (mass flow rate)
rho = 890 kg/m^3 (density of engine oil at 366 K)
A = pi * (0.02 m)^2 / 4 = 0.000314 m^2 (cross-sectional area of the tube)
v = 0.01 / (890 * 0.000314) = 3.79 m/s
The Reynolds number for engine oil is:
Re = (rho * v * D) / mu, where
D = 0.02 m (diameter of the tube)
mu = 0.0003 Pa s (dynamic viscosity of engine oil at 366 K)
Re = (890 * 3.79 * 0.02) / 0.0003 = 4.72e5
Using the Reynolds number, we can estimate the hydrodynamic entry length using the following equation:
L_hyd = 0.05 * Re * D
L_hyd = 0.05 * 4.72e5 * 0.02 = 472 m
For the thermal entry length, we use the following equation:
L_th = 0.05 * Re * Pr * D / Nu
where Pr is the Prandtl number and Nu is the Nusselt number. For engine oil at 366 K, Pr = 130 and we can estimate Nu using the Dittus-Boelter correlation:
Nu = 0.023 * Re^(4/5) * Pr^(0.4)
Nu = 0.023 * (4.72e5)^(4/5) * 130^(0.4) = 2600
L_th = 0.05 * 4.72e5 * 130 * 0.02 / 2600 = 7.2 m
For NaK (22%/78%) at 366 K and a flow rate of 0.01 kg/s, we can calculate the mean velocity as follows:
v = m_dot / (rho * A), where
m_dot = 0.01 kg/s (mass flow rate)
rho = 1100 kg/m^3 (density of NaK at 366 K)
A = pi * (0.02 m)^2 / 4 = 0.000314 m^2 (cross-sectional area of the tube)
v = 0.01 / (1100 * 0.000314) = 2.87 m/s
The Reynolds number for NaK is:
Re = (rho * v * D) / mu, where
D = 0.02 m (diameter of the tube)
mu = 0.0018 Pa s (dynamic viscosity of NaK at 366 K)
Re = (1100 * 2.87 * 0.02) / 0.0018 = 3.48e5
Using the Reynolds number, we can estimate the hydrodynamic entry length using the following equation:
L_hyd = 0.05 * Re * D
L_hyd = 0.05 * 3.48e5 * 0.02 = 348 m
For the thermal entry length, we use the following equation:
L_th = 0.05 * Re * Pr * D / Nu
where Pr is the Prandtl number and Nu is the Nusselt number. For NaK at 366 K, Pr = 0.013 and we can estimate Nu using the Sieder-Tate correlation:
Nu = 0.027 * Re^(4/5) * Pr^(0.43)
Nu = 0.027 * (3.48e5)^(4/5) * 0.013^(0.43) = 67.3
L_th = 0.05 * 3.48e5 * 0.013 * 0.02 / 67.3 = 0.008 m
(b)
For water and engine oil, the mass flow rate is given as 0.01 kg/s, and the mean velocity is given as 0.02 m/s.
For water at 300 K:
rho = 996.6 kg/m^3 (density of water at 300 K)
mu = 0.000547 Pa s (dynamic viscosity of water at 300 K)
Pr = 7.02 (Prandtl number of water at 300 K)
Nu = 0.023 * Re^(4/5) * Pr^(0.4)
Re = (rho * v * D) / mu = (996.6 * 0.02 * 0.02) / 0.000547 = 1445
Nu = 0.023 * Re^(4/5) * Pr^(0.4) = 0.023 * (1445)^(4/5) * 7.02^(0.4) = 426.5
L_hyd = 0.05 * Re * D = 0.05 * 1445 * 0.02 = 14.45 m
L_th = 0.05 * Re * Pr * D / Nu = 0.05 * 1445 * 7.02 * 0.02 / 426.5 = 0.021 m
For water at 400 K:
rho = 958.4 kg/m^3 (density of water at 400 K)
mu = 0.000294 Pa s (dynamic viscosity of water at 400 K)
Pr = 4.41 (Prandtl number of water at 400 K)
Nu = 0.023 * Re^(4/5) * Pr^(0.4)
Step-by-step explanation: