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1 | 1 | # Enthalpy of adsorption and the Clausius-Clapeyron relation |
2 | 2 |
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3 | 3 | ## Enthalpy of adsorption |
4 | | -The energy balance of a simple adsorption process can be written as |
| 4 | +The energy balance in differential form for a simple adsorption process can be written as |
5 | 5 |
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6 | | -$$\frac{\mathrm{d}U}{\mathrm{d}t}=\sum_i\dot{n}_ih_i^\mathrm{b}+\dot{Q}$$ |
| 6 | +$$\mathrm{d}U=h^\mathrm{in}\delta n^\mathrm{in}-h^\mathrm{b}\delta n^\mathrm{out}+\delta Q$$ (eqn:energy_balance) |
7 | 7 |
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8 | | -Here the balance is chosen to only include the fluid in the porous medium. The partial molar enthalpy $h_i^\mathrm{b}$ is the enthalpy of the fluid at the point where it enters or leaves the adsorber at which point it can be considered in a bulk state. The component balance is simply |
| 8 | +Here the balance is chosen to only include the fluid in the porous medium. The molar enthalpy $h^\mathrm{b}$ of the (bulk) fluid that leaves the adsorber is at a state that is in equilibrium with the porous medium. In contrast, the incoming stream can be at any condition. Analogously, the component balance is |
9 | 9 |
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10 | | -$$\frac{\mathrm{d}N_i}{\mathrm{d}t}=\dot{n}_i$$ |
| 10 | +$$\mathrm{d}N_i=x_i^\mathrm{in}\delta n^\mathrm{in}-x_i\delta n^\mathrm{out}$$ (eqn:mass_balance) |
11 | 11 |
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12 | | -and can be used in the energy balance to yield |
| 12 | +The differential of the internal energy can be replaced with the total differential in its variables temperature $T$ and number of particles $N_i$. The volume of the adsorber is fixed and thus not considered as a variable. |
13 | 13 |
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14 | | -$$\dot{Q}=\frac{\mathrm{d}U}{\mathrm{d}t}-\sum_ih_i^\mathrm{b}\frac{\mathrm{d}N_i}{\mathrm{d}t}$$ |
| 14 | +$$\mathrm{d}U=\left(\frac{\partial U}{\partial T}\right)_{N_k}\mathrm{d}T+\sum_i\left(\frac{\partial U}{\partial N_i}\right)_{T,N_j}\mathrm{d}N_i$$ (eqn:U_differential) |
15 | 15 |
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16 | | -or in differential form |
| 16 | +Eqs. {eq}`eqn:energy_balance`, {eq}`eqn:mass_balance` and {eq}`eqn:U_differential` can be combined into an expression for the heat of adsorption $\delta Q$ |
17 | 17 |
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18 | | -$$\delta Q=\mathrm{d}U-\sum_ih_i^\mathrm{b}\mathrm{d}N_i$$ |
| 18 | +$$\delta Q=\left(\frac{\partial U}{\partial T}\right)_{N_k}\mathrm{d}T-\left(h^\mathrm{in}-\sum_ix_i^\mathrm{in}\left(\frac{\partial U}{\partial N_i}\right)_{T,N_j}\right)\delta n^\mathrm{in}+\left(h^\mathrm{b}-\sum_ix_i\left(\frac{\partial U}{\partial N_i}\right)_{T,N_j}\right)\delta n^\mathrm{out}$$ |
19 | 19 |
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20 | | -The expression can be rewritten using the total differential of the internal energy (the volume of the adsorber is fixed and therefore not considered as a variable) |
| 20 | +The heat of adsorption can thus be split into a sensible part that depends on the change in temperature, and a latent part that depends on the change in loading. The expression can be simplified by using the definitions of the isochoric heat capacity $C_v=\left(\frac{\partial U}{\partial T}\right)_{N_k}$ and the **partial molar enthalpy of adsorption** |
21 | 21 |
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22 | | -$$\delta Q=\left(\frac{\partial U}{\partial T}\right)_{N_k}\mathrm{d}T-\sum_i\left(h_i^\mathrm{b}-\left(\frac{\partial U}{\partial N_i}\right)_T\right)\mathrm{d}N_i$$ |
23 | | - |
24 | | -The heat of adsorption $\delta Q$ can thus be split into a sensible part that depends on the change in temperature, and a latent part that depends on the change in loading. The expression can be simplified by using the definitions of the isochoric heat capacity $C_v=\left(\frac{\partial U}{\partial T}\right)_{N_k}$ and the **partial molar enthalpy of adsorption** |
25 | | - |
26 | | -$$\Delta h_i^\mathrm{ads}=h_i^\mathrm{b}-\left(\frac{\partial U}{\partial N_i}\right)_T$$ |
| 22 | +$$\Delta h_i^\mathrm{ads}=h_i^\mathrm{b}-\left(\frac{\partial U}{\partial N_i}\right)_{T,N_j}$$ |
27 | 23 |
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28 | 24 | yielding |
29 | 25 |
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30 | | -$$\delta Q=C_v\mathrm{d}T-\sum_i\Delta h_i^\mathrm{ads}\mathrm{d}N_i$$ |
| 26 | +$$\delta Q=C_v\mathrm{d}T-\sum_ix_i^\mathrm{in}\left(h_i^\mathrm{in}-h_i^\mathrm{b}+\Delta h_i^\mathrm{ads}\right)\delta n^\mathrm{in}+\sum_ix_i\Delta h_i^\mathrm{ads}\delta n^\mathrm{out}$$ |
31 | 27 |
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32 | | -If the composition of the bulk phase is fixed, which can be a fair assumption for an adsorption process but is in general not the case for a desorption process, the heat of adsorption simplifies to |
| 28 | +or |
33 | 29 |
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34 | | -$$\delta Q=C_v\mathrm{d}T-\Delta h^\mathrm{ads}\mathrm{d}N$$ |
| 30 | +$$\delta Q=C_v\mathrm{d}T-\sum_ix_i^\mathrm{in}\left(h_i^\mathrm{in}-h_i^\mathrm{b}+\Delta h_i^\mathrm{ads}\right)\delta n^\mathrm{in}+\Delta h^\mathrm{ads}\delta n^\mathrm{out}$$ |
35 | 31 |
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36 | 32 | with the **enthalpy of adsorption** |
37 | 33 |
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38 | | -$$\Delta h^\mathrm{ads}=\sum_ix_i\Delta h_i^\mathrm{ads}=h^\mathrm{b}-\sum_ix_i\left(\frac{\partial U}{\partial N_i}\right)_T$$ |
| 34 | +$$\Delta h^\mathrm{ads}=\sum_ix_i\Delta h_i^\mathrm{ads}=h^\mathrm{b}-\sum_ix_i\left(\frac{\partial U}{\partial N_i}\right)_{T,N_j}$$ |
| 35 | + |
| 36 | +For **pure components** the balance equations simplify to |
| 37 | + |
| 38 | +$$\delta Q=C_v\mathrm{d}T-\left(h^\mathrm{in}-h^\mathrm{b}\right)\delta n^\mathrm{in}-\Delta h^\mathrm{ads}\mathrm{d}N$$ |
39 | 39 |
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40 | 40 | ## Clausius-Clapeyron relation for porous media |
41 | 41 | The Clausius-Clapeyron relation relates the $p-T$ slope of a pure component phase transition line to the corresponding enthalpy of phase change. For a vapor-liquid phase transition, the exact relation is |
@@ -70,15 +70,15 @@ $$\frac{\mathrm{d}\ln p}{\mathrm{d}\frac{1}{RT}}=-\frac{RT^2}{pv^\mathrm{b}}\lef |
70 | 70 |
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71 | 71 | Here the directional derivative $\frac{\mathrm{d}\mu_i}{\mathrm{d}T}$ could be replaced with a partial derivative amongst the variables describing the adsorbed fluid. The partial derivative can then be replaced using a Maxwell relation based on the Helmholtz energy $F$ as follows |
72 | 72 |
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73 | | -$$\left(\frac{\partial\mu_i}{\partial T}\right)_{N_k}=\left(\frac{\partial^2 F}{\partial T\partial N_i}\right)=-\left(\frac{\partial S}{\partial N_i}\right)_T$$ |
| 73 | +$$\left(\frac{\partial\mu_i}{\partial T}\right)_{N_k}=\left(\frac{\partial^2 F}{\partial T\partial N_i}\right)=-\left(\frac{\partial S}{\partial N_i}\right)_{T,N_j}$$ |
74 | 74 |
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75 | 75 | Using the Maxwell relation together with the compressibility factor of the bulk phase $Z^\mathrm{b}=\frac{pv^\mathrm{b}}{RT}$ in eq. {eq}`eqn:clausius_clapeyron_intermediate` results in |
76 | 76 |
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77 | | -$$\frac{\mathrm{d}\ln p}{\mathrm{d}\frac{1}{RT}}=-\frac{T}{Z^\mathrm{b}}\left(s^\mathrm{b}-\sum_ix_i\left(\frac{\partial S}{\partial N_i}\right)_T\right)$$ |
| 77 | +$$\frac{\mathrm{d}\ln p}{\mathrm{d}\frac{1}{RT}}=-\frac{T}{Z^\mathrm{b}}\left(s^\mathrm{b}-\sum_ix_i\left(\frac{\partial S}{\partial N_i}\right)_{T,N_j}\right)$$ |
78 | 78 |
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79 | 79 | Finally, using $h^\mathrm{b}=Ts^\mathrm{b}+\sum_ix_i\mu_i$ and $\mathrm{d}U=T\mathrm{d}S+\sum_i\mu_i\mathrm{d}N_i$ leads to |
80 | 80 |
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81 | | -$$\frac{\mathrm{d}\ln p}{\mathrm{d}\frac{1}{RT}}=-\frac{1}{Z^\mathrm{b}}\left(h^\mathrm{b}-\sum_ix_i\left(\frac{\partial U}{\partial N_i}\right)_T\right)=-\frac{\Delta h^\mathrm{ads}}{Z^\mathrm{b}}$$ (eqn:deriv_relation_hads) |
| 81 | +$$\frac{\mathrm{d}\ln p}{\mathrm{d}\frac{1}{RT}}=-\frac{1}{Z^\mathrm{b}}\left(h^\mathrm{b}-\sum_ix_i\left(\frac{\partial U}{\partial N_i}\right)_{T,N_j}\right)=-\frac{\Delta h^\mathrm{ads}}{Z^\mathrm{b}}$$ (eqn:deriv_relation_hads) |
82 | 82 |
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83 | 83 | The relation is exact and valid for an arbitrary number of components in the fluid phase. |
84 | 84 |
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