Pipes transporting domestic waste can get clogged due to solid particles slowly accumulating on the inner walls of the pipe. The objective of this question is to build a simple model to estimate how...
Pipes transporting domestic waste can get clogged due to solid particles slowly accumulating on the inner walls of the pipe. The objective of this question is to build a simple model to estimate how clogging affects the magnitude of the primary losses in such a pipe. We consider a cylindrical pipe of length L, initial diameter Do and initial roughness Eo. Waste, assumed to be a Newtonian liquid of density ρ and dynamic viscosity μ, circulates in this pipe at a constant volume flow rate Q. The accumulation of solid material on the inner walls of the pipe is supposed to be homogeneous, causing the pipe diameter to decrease with time t as D(t-Do exp (-t/T) and the pipe roughness to increase with time as ε(t) exp (t/T), where τ>0 is a constant. We further assume that viscous friction in this pipe is dominated by roughness, so that the friction coefficient A follows von Kármán's law: h--20 log(#) Compute the value of the ratio ΔΡ(0) / ΔΡ(T) between the pressure loss ΔΡ(0) at t-0 and the pressure loss ΔΡ(T) at time t-T, in the limit where ε0 / Do << 1. Answer
Pipes transporting domestic waste can get clogged due to solid particles slowly accumulating on the inner walls of the pipe. The objective of this question is to build a simple model to estimate how clogging affects the magnitude of the primary losses in such a pipe. We consider a cylindrical pipe of length L, initial diameter Do and initial roughness Eo. Waste, assumed to be a Newtonian liquid of density ρ and dynamic viscosity μ, circulates in this pipe at a constant volume flow rate Q. The accumulation of solid material on the inner walls of the pipe is supposed to be homogeneous, causing the pipe diameter to decrease with time t as D(t-Do exp (-t/T) and the pipe roughness to increase with time as ε(t) exp (t/T), where τ>0 is a constant. We further assume that viscous friction in this pipe is dominated by roughness, so that the friction coefficient A follows von Kármán's law: h--20 log(#) Compute the value of the ratio ΔΡ(0) / ΔΡ(T) between the pressure loss ΔΡ(0) at t-0 and the pressure loss ΔΡ(T) at time t-T, in the limit where ε0 / Do