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Prolems Solutions

The document contains two problems. The first problem involves developing a mathematical model to describe how pressure varies over time in a gas surge drum. The second problem involves developing a mathematical model using energy balance equations to describe the exit temperature of a heated stream in a stirred tank process where the incoming stream temperature and ambient temperature can vary.

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Mohab Osama
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93 views4 pages

Prolems Solutions

The document contains two problems. The first problem involves developing a mathematical model to describe how pressure varies over time in a gas surge drum. The second problem involves developing a mathematical model using energy balance equations to describe the exit temperature of a heated stream in a stirred tank process where the incoming stream temperature and ambient temperature can vary.

Uploaded by

Mohab Osama
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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Problems

1. Surge drums are often used as intermediate storage for gas streams: that are
transferred between chemical process units. Consider a drum depicted in Figure 1,
where qi is the inlet molar flow rate of gas and q is the outlet molar flow rate of
the gas.
Required:
1. Develop a model that describes how the pressure in the tank varies with time.
2. State your assumptions

qi q
(mole/ time) P (mole/ time)

Figure 1: Gas surge drum

2. A completely enclosed stirred-tank heating process is used to heat an incoming


stream whose flow rate varies. The heating rate from this coil and the volume are
both constant.
Develop a mathematical model (differential and algebraic equations) that
describes the exit temperature if heat losses to the ambient occur and if the
ambient temperature Ta​) and the incoming stream’s temperature (Ti​) both can
vary.
Notes: ρ and Cρ​ are constants.
U, the overall heat transfer coefficient, is constant.
As​ is the surface area for heat losses to ambient.
Ti​>Ta​ (inlet temperature is higher than ambient temperature).

Figure:2
Answer the problem (1)

qi q
(mole/ time) P (mole/ time)

1- Basis:
Total continuity equation

2- Assumptions:
• Isothermal ( no temperature mentioned)
• Constant area (C. S. A)
• Behaves as an ideal gas

3- Mathematical model:
[ mass flow into the system] – [mass flow out of the system]= [time rate of
the change]
Molar flow rate= (mass flow rate/Molecular weight)
∵ n•= [m•/M]

∴m•= n• *M

∵n• =q

∴ m•= q * M

∴qi *M – q*M = dm/dt

∵ m=Ꝭ *V

qi *M – q*M = d(Ꝭ *V)/dt

∵for ideal gas Ꝭ= [P*M/R*T]

qi *M – q*M = d((P*M/R*T)*V)/dt
M*(qi– q)= [V*M/R*T] d(P)/dt ÷M

(qi– q)= [V/R*T] d(P)/dt

(pressure changes with time)

Answer the problem (2)

1- Basis:
Energy balance equation
2- Assumptions:
• Constant Cp
• Neglect kinetic energy
• Neglect potential energy
• Neglect shaft-work
• Perfect mixing
• Constant Ꝭ, V
• No phase change(Liquid)
3- Mathematical model:
w: mass flow rate of feed
Q: heat loss to surrounding is governed by the equation
Q= U*A(T-Ta)

WRITE THE ENERGY EQUATION BALANCE

Temperature change with time?


w*Cp*Ti – w*Cp* T – Q loss + Q coil = d(Ꝭ*V*Cp*T)/dt

∵Q= U*A(T-Ta)

∴w*Cp*Ti – w*Cp* T – U*A(T-Ta)+ Q coil = d(Ꝭ*V*Cp*T)/dt

∴w*Cp*Ti – w*Cp* T – U*A(T-Ta)+ Q coil =( Ꝭ*V*Cp) dT/dt

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