10 Thermodynamic Analysis of Activated Carbon–Ethanol …
187
10.7 Mathematical Modelling
The mathematical model is derived on the basis of simple thermodynamic cycles
described in Fig. 10.1. This model is based on the adsorption equilibrium equations of
the working pair and heat flows. The assumptions made to simplify the calculations
are as follows. (i) Equilibrium condition is reached in adsorbent bed during both
adsorption and desorption process, it implies that the amount of adsorbate adsorbed
or desorbed is calculated using equilibrium adsorption equation. (ii) The pressure,
temperature and concentration of adsorbate are uniform throughout the adsorbent
bed. (iii) No change in the concentration of adsorbate during sensible heating and
cooling of adsorbent bed. (iv) Specific heat of refrigerant is considered to be constant
over the working temperature range. (v) Temperature difference between the heat
transfer fluid and the adsorbent bed is considered to be negligible. (vi) Temperature
difference between the heat transfer fluid and the refrigerant is considered to be
negligible in the evaporator/condenser. And, (vii) system is well insulated i.e., no
heat loss from the system to the environment.
The Dubinin–Astakhov (D-A) equation is used to calculate sorption uptake of
adsorbate in the adsorbent bed as a function of temperature (T ) and pressure (P)
(Tamainot-Telto and Critoph 1997; Critoph and Turner 1988; Saha et al. 2009, 2011).
x(T, T psat ) = x 0 exp
−k
T
T psat
− 1
n
(10.1)
where, x o is the maximum concentration of adsorbate obtained under saturated concentration, k and n are empirical coefficients. Values x o , k, and n depends on the
nature of adsorbent-adsorbate pair and are constants for a given adsorbent-adsorbate
pair. T is the temperature of the bed in Kelvin and T psat is the saturation temperature
(K) of adsorbate at the bed pressure (p).
Heat analysis of basic cycle
Sensible heat (Q sh ) is the energy required to increase the bed temperature from
minimum adsorption temperature T a to minimum desorption temperature T b and
pressure from p e to p c prior to the onset of desorption process (process a-b on
thermodynamic cycle in Fig. 10.1). Q sh heat supply per unit kg of adsorbent is given
by:
Q sh =
T b
T a
c ad (T ) dT +
T b
T a
R m c
re f
ad dT +
T b
T a
c vr x a dT
(10.2)
where, c ad is the specific heat of activated carbon, c
re f
ad is the specific heat of activated
carbon at reference temperature (300 K) and c vr is average constant volume specific
heat of liquid and gaseous refrigerant. R m = (c m m m )/(c
re f
ad m ad ) is defined as the
ratio of heat capacity of the combined container material of the adsorber bed and HTF
187
10.7 Mathematical Modelling
The mathematical model is derived on the basis of simple thermodynamic cycles
described in Fig. 10.1. This model is based on the adsorption equilibrium equations of
the working pair and heat flows. The assumptions made to simplify the calculations
are as follows. (i) Equilibrium condition is reached in adsorbent bed during both
adsorption and desorption process, it implies that the amount of adsorbate adsorbed
or desorbed is calculated using equilibrium adsorption equation. (ii) The pressure,
temperature and concentration of adsorbate are uniform throughout the adsorbent
bed. (iii) No change in the concentration of adsorbate during sensible heating and
cooling of adsorbent bed. (iv) Specific heat of refrigerant is considered to be constant
over the working temperature range. (v) Temperature difference between the heat
transfer fluid and the adsorbent bed is considered to be negligible. (vi) Temperature
difference between the heat transfer fluid and the refrigerant is considered to be
negligible in the evaporator/condenser. And, (vii) system is well insulated i.e., no
heat loss from the system to the environment.
The Dubinin–Astakhov (D-A) equation is used to calculate sorption uptake of
adsorbate in the adsorbent bed as a function of temperature (T ) and pressure (P)
(Tamainot-Telto and Critoph 1997; Critoph and Turner 1988; Saha et al. 2009, 2011).
x(T, T psat ) = x 0 exp
−k
T
T psat
− 1
n
(10.1)
where, x o is the maximum concentration of adsorbate obtained under saturated concentration, k and n are empirical coefficients. Values x o , k, and n depends on the
nature of adsorbent-adsorbate pair and are constants for a given adsorbent-adsorbate
pair. T is the temperature of the bed in Kelvin and T psat is the saturation temperature
(K) of adsorbate at the bed pressure (p).
Heat analysis of basic cycle
Sensible heat (Q sh ) is the energy required to increase the bed temperature from
minimum adsorption temperature T a to minimum desorption temperature T b and
pressure from p e to p c prior to the onset of desorption process (process a-b on
thermodynamic cycle in Fig. 10.1). Q sh heat supply per unit kg of adsorbent is given
by:
Q sh =
T b
T a
c ad (T ) dT +
T b
T a
R m c
re f
ad dT +
T b
T a
c vr x a dT
(10.2)
where, c ad is the specific heat of activated carbon, c
re f
ad is the specific heat of activated
carbon at reference temperature (300 K) and c vr is average constant volume specific
heat of liquid and gaseous refrigerant. R m = (c m m m )/(c
re f
ad m ad ) is defined as the
ratio of heat capacity of the combined container material of the adsorber bed and HTF