In a lot of cases the resistance is also a function of the

Question

Question 1 - The Universal Flow Principle applies to all systems that involve the flow of mass or energy. It is surprising that most of the equations relating to flow can be re-arranged into the form q= ΔF/R (where q = the flow rate of mass or energy, ΔF is the driving force and R is the resistance). In a lot of cases, the resistance is also a function of the driving force itself, as you will see below. Using the Universal Flow Principle find the equation for the flow resistance and define the driving force for the following formulas.

a. Radiant heat transfer equation: q_r = σFEA(T_h⁴ - T_c⁴), where q_r is the heat transfer rate, σ is the Stefan Bollzmann constant, F is the shape factor, E is the emissivity, A is the surface area, T_h is the hot surface temperature and T_c is the cold surface temperature.

b. Pressure loss in a liquid pipeline: ΔP = Kv3γ/2g where ΔP is the pressure drop in the direction of fluid flow, K is an empirical loss coefficient, v is the flow velocity, γ is the liquid specific weight and g is the acceleration due to gravity.

Question 2 - A vertical cylindrical tank is being filled with water, while at the same time water is being drained as shown in Figure below. Provide:

a. A sketch of the analogous flow network using a capacitor symbol to indicate liquid volume storage.

b. Let h = liquid level height; t = time; R = 988.1(h)^0.5; V_i =  0.5+0.5cos(0.05t), the inlet flow rate; D = 2.5, tank diameter;  γ = 60; liquid specific weight; and h₀ = 10, initial h. Assume that the units are consistent and the exit pressure P_e = 0. Set up an appropriate model (differential equation) for simulating h versus t.

Question 3 - A shipping crate (weight = W) is suspended from a long hoisting cable as shown in the figure below. A large temporary displacement of the crate causes pendulum-like movement. The air drag on the crate is F_d = CV_1.5, where C is a constant and v is the tangential velocity. Assume consistent units and that only relatively small displacement takes place, i.e., θ << 1. Since a small displacement takes place, the small angle approximation can be used. Provide:

a. A free body diagram of the problem.

b. Set up an appropriate differential equation model for use in simulating the angular displacement, θ, versus time, t.