Chapter 4 - Conservation of
Energy
OVERVIEW
Conservation of energy is the topic of Chapter 4. The challenge problem explores different types of energy and how they may be harnessed, including renewable biomass resources; worked example problems include enthalpy change during photosynthesis, hydroelectric power, and photosynthesis in green plants. Potential, kinetic and internal energy and enthalpy and their rates are defined and illustrated. The movement of energy as heat and work is discussed. Energy balances are illustrated first for closed and isolated systems through classic thermodynamic examples such as the expansion of a gas. Significant attention is spent developing calculation strategies for changes in enthalpy due to changes in temperature, pressure and phase. With these tools, open, steady-state systems such as heat loss during breathing are illustrated. Enthalpy changes associated with chemical reactions are calculated using heats of formation or combustion. Complete and incomplete respiration in the human body are given as examples. Dynamic systems include the start-up of a bioreactor and the use of basal metabolic rate to estimate weight gain.
EXAMPLE PROBLEMS
_{}
The limit as T_{} 0 becomes
_{}
where the _{}indicates a partial derivative. Partial derivatives are used when a function (_{} in this case) is dependent on more than one variable (T and P in this case). For the purposes of this problem, the partial derivative can be treated like an ordinary derivative.
There is a similar relationship between the heat capacity at constant volume, C_{v}, and the partial derivative of specific internal energy, U, with respect to temperature, as follows:
_{}
From these heat capacity definitions and the definition of enthalpy given in this chapter, derive the relationship between C_{p} and C_{v} for an ideal gas in terms of n (the number of moles), R (ideal gas constant), and other variables that may be necessary.
Given the following thermodynamic data, find the heat of reaction for the forward reaction.
Species |
_{} (kJ/mol)* |
Adenosine triphosphate (ATP) |
-2981.79 |
H_{2}O |
-286.65 |
Adenosine diphosphate (ADP) |
-2000.19 |
P_{i} |
-1299.13 |
where a, b, c, d, e and f are the
stoichiometric coefficients.
Suppose a continuous bioreactor with a
working volume of 100 liter is used to produce the recombinant protein. A
stream of medium containing essential nutrient including glucose and ammonia
flows into the reactor at a rate of 10 l/hr. The medium contains 50 g/L of
glucose and sufficient quantity of ammonia (as calculated later). The exit stream
contains the cells that harbor the recombinant protein. Under these conditions,
only negligible quantity of glucose can be detected in the exit stream (that
is, all glucose can be assumed to be consumed in the reactor). The temperature
of the inlet stream and the bioreactor are set at 25°C. Assume the
reactor is well insulated and the amount of shaft work involved due to mixing
is negligible.
Suppose the reactor has been operated for a
while and is at steady state, do the following:
A.) How much ammonia is required?
B.) What is the recombinant protein production rate?
C.) It is proposed that the heat of reaction
can be related to the oxygen consumption rate by the following expression:
_{}kJ/mol of oxygen consumed
How good is this correlation when compared with that calculated from using the heat of combustion?
D.) What is the heat addition/removal rate in order to maintain the
reactor at 25°C?
E.) In the middle of the run, the heat
exchanger malfunctioned; as a result, no heat can be added or removed from the
reactor. Assume that culture behavior remains the same as before within this
temperature range (this may not be a good assumption). What is the temperature
of the reactor one hour after the mishap? Assume there is no heat loss from the
reactor to the surrounding.
Data:
· Standard
heat of combustion of glucose, _{}= -2,805 kJ/mol
· Standard
heat of combustion of ammonia, _{}= -382.6 kJ/mol
· Standard
heat of combustion of biomass, _{}= -551 kJ/mol
· Standard
heat of combustion of recombinant protein, _{}= -567 kJ/mol
· Assume the specific heat
capacity of medium and broth are similar to that of water = 1 cal/g
· Assume the density of medium
and broth are similar to that of water = 1 g/cm^{3}
Hints:
·
Since the yield of recombinant protein from
glucose is about 20% that for cells, that implies the yield is (0.2)(0.48) =
0.096 g g^{-1} (g of recombinant protein produced per g of glucose
consumed).