Week 10 Flashcards
Conservation of mass in a streamflow model
- Change in mass = mass in - mass out
- Considering a time period /\t and infinitesimal time period…
(density x Qin x dt) - (density x Q x dt) = (density x dV)
- /dt / density
dV/dt = Qin-Q
- /A
dv/dt = qin-q
(where qin=qro-qd)
Mass =
volume x density
v =
V/A
- normalising for the catchment area
q =
Q/A
- normalising for the catchment area
Using q=av^b
- Differentiate q=av^b (use u=v^b)
- Substitute in dv/dt=
- Eliminate v using relationship between v and q
i. e. rearranging q=av^b - alpha = a^(1/b)b
beta = 2- 1/b; 1
What does the master recession curve show?
Spikes = individual rainfall events
After rain stops = recedes = recession event
Plot on log curve = determine end of significant recession events
Linear reservoir equation; considering when b=1
- Beta = 1
dq/dt = alpha(qin-q)
- During recession qin=0
dq/dt= - alphaq
- Integrate w.r.t t
lnq= - alphat + c
- Boundary conditions (q=qo when t=0)
C=lnqo
lnq=lnqo-alphat
= SAME EQUATION AS MASTER RECESSION CURVE
Significance of considering b=1 in the linear reservoir equation
Hydrologists often assume b=1
Therefore
alpha = a, q = av
THEREFORE
q is a linear function of v i.e. linear reservoir equation
Putting Manning’s equation into the form of q=av^b i.e. turbulent flow
- V=AH so v=H
- Ac = Sv and m=v
a = S/An sqrt(-dz/dt) b = 5/3 N.B NOT ONE!!!
Putting Poiseuille’s formula into the form of q=av^b i.e. laminar flow in a pipe
Ac = piD^2/4 S = piD m = Ac/S = D/4
a = (-density x g x S)/(Co x miu) x dh/dx b = 3
Co = shape factor
Putting Poiseuille’s formula into the form of q=av^b i.e. between two parallel planes
Ac = Bw S = 2B m = Ac/S = w/2
a = (-density x g x S)/(Co x miu) x dh/dx b = 3
Co = shape factor
N.B. h so small = negligible
Values of Co for different shapes
Circular pipe = 2
2 // planes = 3
General geometry 2=
Values of a and b in unconfined aquifer flow
a = 64K/(piSyL)^2
b=2
limit of beta (as b tends to infinity) = 2
