WE'VE NOW DERIVED THE ASSIMILATION FACTOR!!! W C= 2+kV+vA c=Lw a assimilation factor =0+kV+vA physics chemistry,biology “transport” kinetics
c W Q kV vA a = assimilation factor = Q + kV + vA physics chemistry, biology WE’VE NOW DERIVED THE ASSIMILATION FACTOR!!! a c W 1 “transport” “kinetics
TEMPORAL ASPECTS OF POLLUTANT REDUCTION Questions: How long will it take for improvement to occur? What will the shape of the recovery look like? W(t) 0
TEMPORAL ASPECTS OF POLLUTANT REDUCTION Questions: • How long will it take for improvement to occur? • What will the shape of the recovery look like? 0 t W(t) 0 t c ?
TIME-VARIABLE SOLUTIONS ANALYTICAL VS.NUMERICAL dc W(t +c= dt analytical (closed form) numerical calculus computer continuous discrete equation table of values
TIME-VARIABLE SOLUTIONS ANALYTICAL VS. NUMERICAL dc dt c W(t) V analytical (closed form) continuous calculus equation numerical discrete computer table of values
IF WE HAVE COMPUTERS: WHY ANALYTICAL SOLUTIONS??? Great for insights and fundamental understanding For example,dimensionless numbers *Quick assessments "Back of the envelope"calculations 举 Useful for checking numerical computer solutions
IF WE HAVE COMPUTERS: WHY ANALYTICAL SOLUTIONS??? Great for insights and fundamental understanding For example, dimensionless numbers Quick assessments “Back of the envelope” calculations Useful for checking numerical computer solutions
EIGENVALUES de =w-Qc-kve Divide by volume: W(t) W(t) λ=eigenvalue=
EIGENVALUES dc dt Q V k c W(t) V ÷ dc dt c W(t) V = eigenvalue = Q V k V dc dt = W Qc kVc Divide by volume: