. A milk storage tank on a dairy farm is equipped with a refrigeration compresso

Question

. A milk storage tank on a dairy farm is equipped with a refrigeration compressor which removes Q (Btumin) ofheat from warm milk. The insulated and perfectly mixed tank is initially filled with Vo(fe) of warm milk at a temperature of To (°F). The compressor is then turned on and begins to chill down the milk at the same time that fresh warm milk at To F) is added at a volumetric flow rate of q (ft/min) through a pipeline from the milking parlor. The total volume of milk in the tank increases to Vr after all of the cows have been milked. (a) Derive a dynamic model for the process so that the temperature of the milk can be determined as a function of time. Perform a degrees of freedom analysis. Identify all of the model constants, input variables and output variables. (b)

Explanation / Answer

1. given milk storage tank

Heat removed per minute = Q Btu / min

initial volume = Vo ft^3 warm milk, To

Milk inflow rate = q ft^3/min

a. heat capacity of milk = c

then

in t minutes

heat removed = Qt

liquid added = qt

at time t the volume is V(t)

then mass = rho*V(t)

from heat balance

rho*V(t)*c*dT/dt = rho*q*c*To - Q

rho*V(t)*c*dT = (rho*q*c*To - Q)dt

also

dV/dt = q

V = qt + C

at t = 0

V = Vo

V = Vo + qt

hence

rho*c*dT = (rho*q*c*To - Q)dt/(Vo + qt)

integrating

T = To - (Q/rho*c - q*To)*ln((1 + qt/Vo))

where c is hjerat capaxcity of milk

b. input variables are t only ( others are already known)

output is temperature wrt time

ther eis one degree of freedom

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