过程控制04

⎡ mass or energy ⎤ ⎡ mass or energy ⎤ = ⎢entering the system ⎥ _ ⎢leaving the system ⎥ ⎢ from t to t + Δt ⎥ ⎢ from t to t + Δt ⎥ ⎣ ⎦ ⎣ ⎦
Integral balances
Component material balances
Let C A and CP represent the molar concentrations of A and P (moles/volume). dVC A (2) = Fi C Ai − FC A + VrA dt dVCP = − FCP + VrP dt where rA and rP represent the rate of generation of species A and P per unit volume.
Material and energy balances
Many chemical processes have important thermal effects. Developing correct energy balance equations is not trivial.
C Ai − C A X= C Ai so, C Ai − C A (kV / Fs ) X= = C Ai (kV / Fs ) + 1
The conversion is a function of kV / Fs , which is known as the Damkohler number (丹姆克尔数). Two different chemical-reaction systems can have the same conversion if their Damkohler numbers are the same.
dt
= qi − q
PV = nRT ,
dn d ( PV / RT ) V dP = = dt dt RT dt
？
dP RT = (qi − q)ystem!
Outlet flow as a function of gas drum pressure
Consider the case where the outlet molar flow rate is proportional (正比) to the difference in gas-drum pressure and the pressure in the downstream header piping, Ph.
No single model of a process exists!
Lumped parameter (集中参数) system models
These models consist of initial-value ordinary differential equations (ODE常 微 分 方 程 ), often based on a perfect mixing assumption.
M
t +Δt
−M t =
t +Δt •
∫
t
min dt −
t +Δt •
∫
t
t +Δt
mout dt =
∫
t
(min − mout )dt
•
•
• dM • = min − mout dt
Representing M = V ρ , min as Fin ρin and mout as Fout ρ dV ρ = Fin ρin − Fout ρ dt Note that the system is perfectly mixed, ρ out = ρ
？
β
A self-regulating system!
An lsothermal (绝热) chemical reactor example
Fi CAi F CA CB CP
Ethylene oxide (环氧乙烷,A) is reacted with water (B ) in continuously stirred tank reactor (连续搅拌化学反应 器,CSTR) to form ethylene glycol (P,乙二醇). A+ B → P dV = Fi − F (1) dt
•
•
Instantaneous balances
⎡ rate of mass ⎤ ⎢accumulation ⎥ ⎣ ⎦
= ⎡ rate of mass ⎤ _ ⎡ rate of mass ⎤ ⎢ in by flow ⎥ ⎢ out by flow ⎥ ⎣ ⎦ ⎣ ⎦
• dM • = min − mout dt
Steady state
• dM • = min − mout = 0 dt
min = mout or Fin ρin = Fout ρ
Steady-state (稳态) relationships are often used for process design and determination of optimal operating conditions.
dV ρ = Fin ρin − Fout ρ dt
This is the same result obtained using an integral balance. The integral balance method takes longer time to arrive at the same result, but it is probably clearer when developing distributed parameter (分布参数) (partial differential equation-based 基于偏微分方程) models.
•
•
Gas surge drum (气体缓冲罐) example
Surge drums are often used as intermediate storage (中间 储藏) capacity for gas streams that are transferred between chemical process units.
Concept of conversion
The concept of conversion is important in chemical-reaction engineering. The conversion of reactant A is defined as the fraction of the feed-stream component that is reacted.
Reasons for modeling
Improving or understanding industrial process operation is a major overall objective. Models are often used for
(i) operator training. Process operators can learn the proper response to upset conditions, before having to experience them on the actual process. (ii) process design. Can be used to properly design industrial process equipment for a desired production rate. (iii) safety system analysis, or (iv) process control.
CHAPTER 2 : Fundamental models
When I complete this chapter, I want to be able to do the following.
•Write balance equations using the integral (积分) or instantaneous (瞬时) methods •Determine the state, input, and output variables and the parameters for a particular model (set of equations) •Determine the necessary information to solve a system of dynamic equations •Linearize a set of nonlinear equations to find the state space model
x = f ( x, u , p ) y = g ( x, u , p )
•
Integral balances
An integral balance is developed by viewing a system at two different “snapshots” in time.
mass or ⎡ ⎤ ⎡ mass or ⎤ ⎢energy inside the ⎥ − ⎢energy inside the ⎥ ⎢system at t + Δt ⎥ ⎢ system at t ⎥ ⎣ ⎦ ⎣ ⎦
Steady-state solution
dC A dCP = =0 Let dt dt ( Fs / V )C Ais C Ais , = C As = ( Fs / V ) + k (kV / Fs ) + 1 (kV / Fs )C Ais kC Ais = CPs = ( Fs / V ) + k (kV / Fs ) + 1 Notice that the concentrations are a function of the space velocity (空速, Fs / V ), which has units of inverse time.