Kerosene of relative density \(0.82\) and kinematic viscosity \(2.3 \mathrm{~mm}^{2} \cdot \mathrm{s}^{-1}\) is to be pumped through \(185 \mathrm{~m}\) of galvanized iron pipe \((k=0.15 \mathrm{~mm})\) at \(40 \mathrm{~L} \cdot \mathrm{s}^{-1}\) into a storage tank. The pressure at the inlet end of the pipe is \(370 \mathrm{kPa}\) and the liquid level in the storage tank is \(20 \mathrm{~m}\) above that of the pump. Neglecting losses other than those due to pipe friction determine the size of pipe necessary.

Dynamic viscosity (\textbackslash(mu)) is a property of a fluid that measures its resistance to gradual deformation by shear or tensile stress. In simpler terms, it's a measure of how 'thick' or 'sticky' a fluid is. The dynamic viscosity can be calculated using the kinematic viscosity (\textbackslash(nu)) and the density (\textbackslash(rho)) of the fluid. The formula is: \[ \mu = u \times \rho \] where: - \(u = 2.3 \times 10^{-6} \text{ m}^2 \cdot \text{s}^{-1} \) is the kinematic viscosity - \(\rho = 820 \text{ kg } \cdot \text{ m }^{-3} \) is the density of kerosene. Therefore, \( \mu = 2.3 \times 10^{-6} \text{ m}^2 \cdot \text{s}^{-1} \times 820 \text{ kg } \cdot \text{ m }^{-3} = 1.886 \times 10^{-3} \text{ Pa } \cdot \text{ s} \). Dynamic viscosity is crucial for pipe flow calculations because it influences the Reynold's number, which dictates whether the flow is laminar or turbulent.

Flow velocity (\textbackslash(v)) is the speed at which the fluid flows through a pipe. It can be calculated using the flow rate (\textbackslash(Q)) and the cross-sectional area of the pipe (\textbackslash(A)). The continuity equation, which states that the flow rate is the product of the flow velocity and the cross-sectional area, is given by: \[ Q = v \times A \] The cross-sectional area of a circular pipe with diameter \(d\) is given by \(A = \pi \left( \frac{d}{2} \right)^2 = \frac{\pi d^2}{4} \). Therefore, the flow velocity can be given by: \[ v = \frac{4Q}{\pi d^2} \] Given the flow rate, \( Q = 40 \text{ L/s} = 0.04 \text{ m}^3 \cdot \text{s}^{-1} \), the velocity in terms of diameter is: \[ v = \frac{0.16}{\pi d^2} \] Flow velocity is important because it determines the kinetic energy of the fluid and influences the Reynolds number.

The Reynolds number (\textbackslash(Re)) is a dimensionless quantity used to predict flow patterns in different fluid flow situations. It is calculated using the formula: \[ Re = \frac{\rho v d}{\mu} \] where \(\rho\) is the fluid density, \(v\) is the flow velocity, \(d\) is the pipe diameter, and \(\mu\) is the dynamic viscosity. Substituting the values, we get: \[ Re = \frac{820 \times 0.16 d}{1.886 \times \pi d^2} = \frac{69.56}{\pi d} \] The Reynolds number helps in determining whether the flow is laminar (\textbackslash(Re < 2000)), transitional (2000 < \textbackslash(Re < 4000)), or turbulent (\textbackslash(Re > 4000)). In this exercise, assuming a turbulent flow helps in using the appropriate friction factor correlation later.

The Darcy-Weisbach equation is used to calculate the head loss (\textbackslash(h_f)) due to friction in a pipe. The formula is: \[ h_f = f \frac{L}{d} \frac{v^2}{2g} \] where \(f\) is the friction factor, \(L\) is the length of the pipe, \(d\) is the diameter of the pipe, \(v\) is the flow velocity, and \(g\) is the acceleration due to gravity. Applying the given values and assuming an iterative solution to calculate the friction factor individually provides the final pipe diameter. Considering pressure, elevation, and disregarding other losses, the hydraulic head is: \[ \Delta H = \frac{p}{\rho g} + z = 66.019 \text{ m}\] Ensuring that only pipe friction losses impact the head.

The friction factor (\textbackslash(f)) is a dimensionless quantity used in the Darcy-Weisbach equation to account for the resistance along a pipe's wall. For turbulent flow, the friction factor can be estimated using empirical correlations like the Colebrook-White equation, which implicitly relates \(f\) to the Reynolds number and the relative roughness of the pipe: \[ \frac{1}{\sqrt{f}} = -2 \log \left( \frac{\epsilon/D}{3.7} + \frac{2.51}{Re \sqrt{f}} \right) \] where \(\epsilon\) is the roughness height, and \(D\) is the pipe diameter. Due to the iterative nature of solving this equation, flow assumptions and refinement through iteration are used to reach a stable \(f\), ensuring accurate head loss predictions. Friction factor's accurate determination is key to designing efficient piping systems.