A temperature-controlled chamber is shown in Figure. The air temperature inside the chamber is…

A temperature-controlled chamber is shown in Figure.

The air temperature inside the chamber is assumed to be the
same everywhere, namely, T(t). The chamber walls are insulated to reduce heat
loss or gain with its surroundings. Temperature control is achieved by
circulating hot or cold water through pipes located inside the chamber. Heat
exchange occurs between the air inside the chamber and the circulating water in
the pipes. The heat flow from the circulating hot water is Qh(t), and
Qc(t) is the heat flow to the cold water. Heat exchange Q0(t)
also occurs between the air inside
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 A temperature-controlled chamber is shown in Figure.

The air temperature inside the chamber is assumed to be the
same everywhere, namely, T(t). The chamber walls are insulated to reduce heat
loss or gain with its surroundings. Temperature control is achieved by
circulating hot or cold water through pipes located inside the chamber. Heat
exchange occurs between the air inside the chamber and the circulating water in
the pipes. The heat flow from the circulating hot water is Qh(t), and
Qc(t) is the heat flow to the cold water. Heat exchange Q0(t)
also occurs between the air inside and outside the chamber. Ambient temperature
outside the chamber is denoted T0(t). A suitable model for this
thermal system is based on the conservation of energy. CAV=Qh-Qc-Q0 .
V is the volume (ft3) of air in the chamber, and cA is
the thermal capacitance of air (0.01375 Btu/°F/ft3). The heat flow
terms on the right-hand side are given by

Where, h and c are the mass flow rates
(lb/min) of the hot and cold water cp is the specific heat of water
(1 Btu/lb/°F) R is the thermal resistance (°F/Btu/min) of the chamber walls.
The expressions for Qh and Qc assume that the flow rates
of the circulating fluids are great enough that both fluids exit at the same
temperature at which they entered the chamber.

a.       Express
the mathematical model in the form of a differential equation relating the
output T and its derivative to the inputs Th, Tc, and T0.

b.      Find
the time constant and the three steady-state gains of the system. Check the
units to verify that the time constant is in minutes and the steady-state gains
are dimensionless (°F/°F).

c.       Show
that the air temperatures inside and outside the chamber eventually equalize
after both the hot and cold circulating water flows are turned off.

d.      Suppose
the chamber air temperature is required to be higher than the outside ambient
air temperature, which remains constant, that is, T0(t) =T0,
t _ 0. The hot water temperature entering the chamber is three times
greater than the ambient temperature. The initial air temperature inside
the chamber is the same as the outside ambient temperature. Find the analytical
solution for T(t), t _ 0, the air temperature inside the chamber.

e.      Graph
the solution for T(t), t _ 0 in part (d) using the following values:
V=5000 ft3, R=0.25°F/Btu/min,
h=500 lb/min, and 0=60°F.

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