Page 254 - Modelling in Transport Phenomena A Conceptual Approach

234 CHAPTER 7. UNSTEADY-STATE MACROSCOPIC BALANCES

that the variation in the liquid volume as a function of time is given by

where Vgf is the volume of the cylinder at t = 0.
b) If the pump operates at a temperature different from the reference temperature,
show that the mass flow rate provided by the pump is given by

where p and V are the density and the volume of the pump liquid at temperature
T, respectively. Expand p and V in a Taylor series in T about the reference
temperature Tref and show that

where p, the coefficient of volume expansion, is defined by

in which the subscripts L and C represent the liquid and the cylinder, respectively.
Indicate the assumptions involved in the derivation of Eq. (4).

c) Show that the substitution of Eq. (4) into Eq. (3) and making use of Eqs. (1)
and (2) gives the fractional error in mass flow rate as

where

Note that the first and the second terms on the right-side of Eq. (6) represent,
respectively, the steady-state and the unsteady-state contributions to the error
term.
d) Assume that at any instant the temperature of the pump liquid is uniform and
equal to that of the surrounding fluid, i.e., the cylinder wall is diathermal, and
determine the fractional error in mass flow rate for the following cases:

0 The temperature of the fluid surrounding the pump, Tf, is constant. Take
,Bc = 4 x K-l, pL = 1.1 x lod3 K-' , and Tf - TTef = 5K.
The temperature of the surrounding fluid changes at a constant rate of 1 K/ h.
Take Vzf = 500 cm3 and Ro = 25 cm3/ h.