**2.1 Configuration**

222 Heat Exchangers – Basics Design Applications

a compact heat exchange reformer replace the heat exchanger and the pre-reformer. The new fuel cell system is illustrated in Fig. 2. The offset strip fin heat exchanger and prereformer are combined into the heat exchange reformer. In this device with the counter-flow type, the high temperature waste gas from the fuel cell flows in the hot passage, and the fuel flows in the cold passage. In particular, the Ni catalyst is coated on the fuel passage surface [6, 7]. When the fuel flows along the passage, the endothermic steam reforming reaction will

take place using the heat transferring from the hot side.

Fig. 1. Schematic view of the traditional SOFC/GT hybrid system.

reformer.

Fig. 2. Schematic view of the SOFC/GT hybrid system with novel concept heat exchange

Several kinds of compact heat exchange reformers have been investigated and designed in the past. In 2001, Kawasaki Heavy Industries in Japan developed a plate-fin heat-exchange reformer with highly dispersed catalyst [8]. A planar micro-channel concept was proposed by Pacific Northwest National Laboratories (PNNL), but this kind of micro-channel device is oriented toward the low to medium power range (20-500W) for man-portable applications The configuration of the heat exchange reformer is similar to the compact heat exchanger. The only difference is that the catalyst is coated in the cold passage to make steam reforming reactions take place.

As shown in Fig. 3, the configuration of the offset strip fin heat exchanger is adopted here. The fin surface is broken into a number of smaller sections. Generally, each type of fin is characterized by its width *X*, height *Y*, thickness *t*, and length of the offset strip fin *l*. The detailed configuration can also be found in other references for the heat exchanger [14-18].

f,h

h h

h h *U*

*f* 

h

4

*h <sup>L</sup> G G P f <sup>K</sup> D*

*L A*

d dx *P U A* 

The fanning friction factor *f* has been developed by many authors. Basing on the data of

0.7422 0.1856 0.3053 0.2659

 

0.1 4.429 0.920 3.767 0.236

(13)

*m*

Finally, the fin efficiency can be simplified by:

transfer coefficient between the fin and the flow.

exit, and turning loss [15], can be expressed by:

where,

**2.3 Pressure loss** 

where, *G u <sup>m</sup>*

Then,

.

Let the friction resistance <sup>1</sup> <sup>2</sup>

2 *f u* ,

Kays & London [14], Manglik & Bergles [17] recommend:

*f*

9.6243Re


1 Re

h h

(8)

(9)

h h tanh( ) *m k m k*

, hh h *kY t* <sup>2</sup> .

h 0,h f,h h h 1 (1 ) *<sup>Y</sup> X Y*

The fin efficiency is mainly influenced by the material, configuration of the fin, and the heat

The frictional pressure loss across an offset strip fin passage and at any associated entry,

 

2 2

(10)

(11)

(12)

2 2 *m m*

Here, turning losses are neglected, so the pressure loss per unit length can be expressed by:

 

> 1 <sup>2</sup> 2 *P U <sup>f</sup> <sup>u</sup>*

Fig. 3. Flow (a) and fin structure (b) diagram of heat exchange reformer.

Taking the hot passage as an example, the calculations for individual geometry variables are listed as following:

$$\text{Passage number: } n\_{\text{h}} = \text{V} / (X\_{\text{h}} + t\_{\text{h}}) \tag{1}$$

$$\text{Offset} \gets \text{strip number: } n\_{\text{hl}} = \text{L} / l\_{\text{h}} \tag{2}$$

$$\text{Cross area of flow passage: } A\_{\text{h}} = \eta\_{\text{h}} X\_{\text{h}} Y\_{\text{h}} \tag{3}$$

$$\text{Heat transfer surface of flow passage: } S\_{\text{h}} = 2\eta\_{\text{h}}(X\_{\text{h}} + Y\_{\text{h}})L + \eta\_{\text{h}}\eta\_{\text{h}}(X\_{\text{h}} + Y\_{\text{h}} + t\_{\text{h}})t\_{\text{h}} \tag{4}$$

$$\text{Wet perimeter: } \mathcal{U}\_{\mathbf{h}} = 4\mathcal{S}\_{\mathbf{h}}/L \tag{5}$$

$$\text{Hydroraulic diameter: } D\_{\text{h}} = 4A\_{\text{h}} \,/\,\text{U}\_{\text{h}} \tag{6}$$

#### **2.2 Passage fin efficiency**

The passage fin efficiency 0 is given by Rosehnow et al. [18] as

$$\eta\_0 = 1 - \frac{S\_\text{f}}{S}(1 - \eta\_\text{f}) \tag{7}$$

where the secondary heat transfer area of a stream *S*f for the hot passage equals *S*h. The total area of the heat exchanger *S* is calculated by the sum of the primary heat transfer surface and the secondary heat transfer area of a stream.

According to Rosehnow et al. [15, 18], the fin efficiency for the offset strip fin with a rectangular section can be approximated by:

$$\eta\_{\rm f,h} = \frac{\tanh(m\_{\rm h} k\_{\rm h})}{m\_{\rm h} k\_{\rm h}} \tag{8}$$

where,

224 Heat Exchangers – Basics Design Applications

Fig. 3. Flow (a) and fin structure (b) diagram of heat exchange reformer.

listed as following:

**2.2 Passage fin efficiency**  The passage fin efficiency

and the secondary heat transfer area of a stream.

rectangular section can be approximated by:

Taking the hot passage as an example, the calculations for individual geometry variables are

Heat transfer surface of flow passage: *S n X Y L nn X Y t t* h h h h hh h h h h 2 *<sup>l</sup>* (4)

0 is given by Rosehnow et al. [18] as

where the secondary heat transfer area of a stream *S*f for the hot passage equals *S*h. The total area of the heat exchanger *S* is calculated by the sum of the primary heat transfer surface

According to Rosehnow et al. [15, 18], the fin efficiency for the offset strip fin with a

 <sup>f</sup> 0 f 1 1 *<sup>S</sup> S*

(b)

Passage number: *n WX t* h hh (1)

Offset strip number: *n Ll* h h *<sup>l</sup>* (2)

Wet perimeter: h h *U SL* 4 (5)

(7)

Hydraulic diameter: h hh *D AU* 4 / (6)

Cross area of flow passage: *A nXY* h h hh (3)

$$m\_{\rm h} = \sqrt{\frac{\alpha\_{\rm h} U\_{\rm h}}{\mathcal{A}\_{\rm h} f\_{\rm h}}} \; \! \; k\_{\rm h} = Y\_{\rm h} / 2 - t\_{\rm h} \; .$$

Finally, the fin efficiency can be simplified by:

$$\eta\_{0,\rm h} = 1 - \frac{Y\_{\rm h}}{X\_{\rm h} + Y\_{\rm h}} (1 - \eta\_{\rm f,h}) \tag{9}$$

The fin efficiency is mainly influenced by the material, configuration of the fin, and the heat transfer coefficient between the fin and the flow.

#### **2.3 Pressure loss**

The frictional pressure loss across an offset strip fin passage and at any associated entry, exit, and turning loss [15], can be expressed by:

$$
\Delta P = \mathbf{4} \cdot f\left(\frac{L}{D\_h}\right) \left(\frac{G\_m}{2\rho}\right) + K \left(\frac{G\_m}{2\rho}\right) \tag{10}
$$

where, *G u <sup>m</sup>* .

Here, turning losses are neglected, so the pressure loss per unit length can be expressed by:

$$\frac{\Delta P}{L} = \frac{\mathcal{U}}{A} \left(\frac{1}{2} f \rho u^2\right) \tag{11}$$

Let the friction resistance <sup>1</sup> <sup>2</sup> 2 *f u* ,

Then,

$$\frac{\text{d}P}{\text{d}\infty} = \frac{\text{U}\sigma}{A} \tag{12}$$

The fanning friction factor *f* has been developed by many authors. Basing on the data of Kays & London [14], Manglik & Bergles [17] recommend:

$$f = 9.6243 \,\mathrm{Re}^{-0.7422} \,\alpha^{-0.1856} \,\delta^{0.3053} \,\gamma^{-0.2659}$$

$$\times \left[1 + 7.669 \times 10^{8} \,\mathrm{Re}^{4.429} \,\alpha^{0.920} \,\delta^{3.767} \,\gamma^{0.236}\right]^{0.1} \tag{13}$$

3. The parameters are considered to be uniform over a cross-section, one dimensional flow

In the cold fuel passage, the chemical species are CH4, H2, CO, CO2, and H2O. Species mass

The mass, momentum, and energy conservation equations for the hot passage and cold passage are established in Table 2 and Table 3, respectively. In the hot passage, the heat transfer to the solid structure is considered. Due to the very thin catalyst coat, the enthalpy changes of the reactions (I-III) are also considered in the cold passage, in addition to the heat

> h hh ( ) *u t x*

 

2 hh hh h h h

 

h h w h hh ( ) *T T <sup>S</sup> u T T*

()( ) *u u PU t x xA*

h h h h 0,h

*t x Cp A L*

2

c c w

 

*t x Cp A L Cp Y*

 

c cc ( ) *u t x* 

 

cc cc c c c

c cc c cc , ,

 

*T T <sup>S</sup> <sup>u</sup> T T H R*

For the solid structures, such as the fins and the separators, the temperature is considered to

() () *T T S S K TT TT*

2 w h w c w w w w

 

(29)

(28)

( )( ) *u u PU t x xA*

be uniform at the same cross-section. The energy conservation equation is written as:

w w h h 0,h c c 0,c

*t x M Cp M Cp*

*<sup>i</sup>* CH ,H ,CO,CO ,H O 42 22 (22)

(23)

(26)

(24)

h

(25)

c

(27)

*k I II III*

 

<sup>1</sup> ( ) *<sup>k</sup> <sup>k</sup>*

 

2. The viscosity dissipation effects are neglected;

balances in the cold fuel passage are considered.

c, c,

transferred from the solid structure.

Momentum conservation equation

Energy conservation equation

Mass conservation equation

Momentum conservation equation

Energy conservation equation

Mass conservation equation

along the passage, without inside circumfluence;

4. For the horizontal fluid, the effect of height change can be omitted.

I , II , III c

*ik k*

c ,

*k C C <sup>u</sup> v R tx Y*

Table 2. Hot passage dynamic mathematical model.

c c c c 0,c

Table 3. Cold passage dynamic mathematical model.

2

*i i* 1

#### **2.4 Heat transfer coefficient**

Generally, the heat transfer coefficient is related to the Colburn factor [15, 17, 18] and is expressed as:

$$\alpha = \text{J} \mathbf{G}\_m \mathbf{c}\_p \text{Pr}^{-2/3} \tag{14}$$

where the Colburn factor 2/3 *J St* Pr and the Prandtl number Pr *<sup>p</sup> c* .

The correlation developed by Manglik & Bergles [17] from the data of Kays & London [14] reads:

$$J = 0.6522 \,\text{Re}^{-0.5403} \,\alpha^{-0.1541} \,\delta^{0.1499} \,\gamma^{-0.0678} \times \left[1 + 5.269 \times 10^{-5} \,\text{Re}^{1.340} \,\alpha^{0.594} \,\delta^{0.456} \,\gamma^{-1.055}\right]^{0.1} \tag{15}$$

#### **2.5 Steam reforming**

In the cold fuel passage, the steam reforming reaction (I), water gas shift reaction (II), and CO2 direct reforming reactions of methane (III) are carried out over a Ni catalyst coat on the passage surface at sufficiently high temperatures, typically above 773K.

Kinetic rate equations for the reactions (I-III) are adopted from Xu and Froment [19]. The three kinetic rate equations are listed in Table 1 as well.

$$\begin{array}{ll} \text{(I)} & \text{CH}\_4\text{+H}\_2\text{O} \Leftrightarrow \text{CO}\text{+3H}\_2\\ \text{(I)} & & p\_{\text{(I)}} = \frac{k\_1}{p\_{\text{H}\_2}^{2.5}} \left( p\_{\text{CH}\_4} p\_{\text{H}\_2\text{O}} - \frac{p\_{\text{H}\_2}^3 p\_{\text{CO}}}{K\_{\text{c1}}} \right) \ast \frac{1}{D \text{EN}^2} \end{array} \tag{16}$$

$$\text{(II)}\tag{\text{II}}\tag{\text{II}}\tag{\text{i}}\tag{\text{}}\tag{\text{i}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\}\tag{\text{}}\tag{\text{}}\tag{\text{}}\tag{\text{}}\}\tag{\text{}}\tag{\text{}}\tag{\text{}}\}\tag{\text{}}\tag{\text{}}\tag{\text{}}\}\tag{\text{}}\tag{\text{}}\}\tag{\text{}}\tag{$$

$$\begin{array}{ll} \text{(III)} & \mathbf{CH\_4} \star \mathbf{H\_2O} \Leftrightarrow \mathbf{CO\_2} \star \mathbf{H\_2} & & \\ & & \mathbf{H\_{(II)}} = \frac{k\_3}{p\_{1\downarrow}^{\rm 3}} \left( p\_{\text{CH}} p\_{1\downarrow}^2 - \frac{p\_{1\downarrow}^4 p\_{\text{CO}\_2}}{K\_\odot} \right) \ast \frac{1}{\text{DEN}^2} \end{array} \tag{18}$$

Table 1. Reaction and its rate in the heat exchange reformer (Xu and Froment, [19]).

The enthalpy changes of chemical reactions are calculated according to Smit et.al [20].

$$\Delta H\_{\text{(I)}} = \Delta F\_{\text{(I)}}^0 - 16373.61 + R \left( 7.951 T\_{\text{c}} - 4.254 e - 3 T\_{\text{c}}^2 + 0.7213 e - 6 T\_{\text{c}}^3 - 0.097 e 5 f T\_{\text{c}} \right) \tag{19}$$

$$
\Delta H\_{\text{(II)}} = \Delta H\_{\text{(II)}}^0 - 7756.56 + R \left( 1.86 T\_\text{c} - 0.27 e - 3 T\_\text{c}^2 + 1.164 e 5 f T\_\text{c} \right) \tag{20}
$$

$$
\Delta H\_{\text{(III)}} = \Delta H\_{\text{(III)}}^0 - 26125.07 + R \left( 10.657 T\_{\text{c}} - 4.624 c - \text{\textdegree{}T}\_{\text{c}}^2 + 0.7213 c - 6 T\_{\text{c}}^3 + 1.067 c \text{\textdegree{}^{\text{f}}T\_{\text{c}}} \right) \tag{21}
$$

#### **3. Mathematic model of heat exchange reformer**

To simplify the complexity of the mathematical model, some assumptions [4, 21] adopted in the theoretic analysis are presented as follows:

1. The heat exchange reformer is adiabatic to the surrounding;


In the cold fuel passage, the chemical species are CH4, H2, CO, CO2, and H2O. Species mass balances in the cold fuel passage are considered.

$$\frac{\partial \mathbf{C}\_{\mathbf{c},j}}{\partial t} = -\mu\_{\mathbf{c}} \frac{\partial \mathbf{C}\_{\mathbf{c},j}}{\partial \mathbf{x}} + \sum\_{k \in \{\text{(I)}, \{\text{(II)}, \{\text{(II)}\}\}} v\_{i,k} \mathbf{R}\_{k} \frac{1}{Y\_{\mathbf{c}}} \qquad i \in \{\text{CH}\_{4}, \text{H}\_{2}, \text{CO,CO}\_{2}, \text{H}\_{2}\text{O}\} \tag{22}$$

The mass, momentum, and energy conservation equations for the hot passage and cold passage are established in Table 2 and Table 3, respectively. In the hot passage, the heat transfer to the solid structure is considered. Due to the very thin catalyst coat, the enthalpy changes of the reactions (I-III) are also considered in the cold passage, in addition to the heat transferred from the solid structure.

Mass conservation equation

226 Heat Exchangers – Basics Design Applications

The correlation developed by Manglik & Bergles [17] from the data of Kays & London [14]

0.1 0.5403 0.1541 0.1499 0.0678 1.340 0.504 0.456 1.055 *<sup>J</sup>* 0.6522Re

In the cold fuel passage, the steam reforming reaction (I), water gas shift reaction (II), and CO2 direct reforming reactions of methane (III) are carried out over a Ni catalyst coat on the

Kinetic rate equations for the reactions (I-III) are adopted from Xu and Froment [19]. The

1 Re

(15)

2/3


4 2

2

4 2

*<sup>k</sup> p p R pp*

2 H CO II CO H O <sup>2</sup> H 2

*<sup>k</sup> p p R pp p K DEN* 

*<sup>k</sup> p p R pp*

1 H CO I 2.5 CH H O 2 H 1

3

*p K DEN* 

*e*

*e*

*e*

4 3 2 H CO III 3.5 CH H O 2 H 3

*p K DEN* 

1

1

1

(16)

(17)

(18)

2

2

2

I I c c cc *H H R T e T e T eT* 16373.61 7.951 4.354 3 0.7213 6 0.097 5 (19)

II II cc c *H H* 7756.56 1.86 0.27 3 1.164 5 *R T e T eT* (20)

III III c c cc *H H* 26125.07 10.657 4.624 3 0.7213 6 1.067 5 *R T e T e T eT* (21)

is related to the Colburn factor [15, 17, 18] and is

*JG c m p* Pr (14)

*c* .

where the Colburn factor 2/3 *J St* Pr and the Prandtl number Pr *<sup>p</sup>*

passage surface at sufficiently high temperatures, typically above 773K.

(I) CH +H O CO+3H 4 2 <sup>2</sup> <sup>2</sup>

(II) CO+H O CO +H 2 22 2 2

(III) CH +2H O CO +4H 42 22 2 2

Table 1. Reaction and its rate in the heat exchange reformer (Xu and Froment, [19]).

The enthalpy changes of chemical reactions are calculated according to Smit et.al [20].

0 2 <sup>3</sup>

0 2

0 2 <sup>3</sup>

To simplify the complexity of the mathematical model, some assumptions [4, 21] adopted in

**2.4 Heat transfer coefficient** 

expressed as:

reads:

**2.5 Steam reforming** 

Generally, the heat transfer coefficient

three kinetic rate equations are listed in Table 1 as well.

**3. Mathematic model of heat exchange reformer** 

1. The heat exchange reformer is adiabatic to the surrounding;

the theoretic analysis are presented as follows:

 

$$\frac{\partial \rho\_{\mathbf{h}}}{\partial t} = -\frac{\partial (\rho\_{\mathbf{h}} u\_{\mathbf{h}})}{\partial \mathbf{x}}\tag{23}$$

Momentum conservation equation

$$\frac{\partial(\rho\_{\text{h}}u\_{\text{h}})}{\partial t} = -\frac{\partial(\rho\_{\text{h}}u\_{\text{h}}{}^{2})}{\partial \mathbf{x}} - \frac{\partial P\_{\text{h}}}{\partial \mathbf{x}} - \frac{\mathcal{U}\_{\text{h}}\sigma\_{\text{h}}}{A\_{\text{h}}} \tag{24}$$

Energy conservation equation

$$\frac{\partial T\_{\rm h}}{\partial t} = -\mu\_{\rm h} \frac{\partial T\_{\rm h}}{\partial \mathbf{x}} - \frac{S\_{\rm h} \alpha\_{\rm h} \eta\_{0, \rm h}}{\rho\_{\rm h} \mathbb{C} p\_{\rm h} A\_{\rm h} L} (T\_{\rm h} - T\_{\rm w}) \tag{25}$$

Table 2. Hot passage dynamic mathematical model.

Mass conservation equation

$$\frac{\partial \rho\_c}{\partial t} = \frac{\partial (\rho\_c \mu\_c)}{\partial x} \tag{26}$$

Momentum conservation equation

$$\frac{\partial(\rho\_c \mu\_c)}{\partial t} = \frac{\partial(\rho\_c \mu\_c^{-2})}{\partial \mathbf{x}} + \frac{\partial P\_c}{\partial \mathbf{x}} - \frac{\mathcal{U}\_c \sigma\_c}{A\_c} \tag{27}$$

Energy conservation equation

$$\frac{\partial T\_{\text{c}}}{\partial t} = \mu\_{\text{c}} \frac{\partial T\_{\text{c}}}{\partial \mathbf{x}} - \frac{S\_{\text{c}} \mu\_{\text{c}} \eta\_{0, \text{c}}}{\rho\_{\text{c}} \text{C} p\_{\text{c}} A\_{\text{c}} L} (T\_{\text{c}} - T\_{\text{w}}) + \frac{1}{\rho\_{\text{c}} \text{C} p\_{\text{c}} Y\_{\text{c}}} \sum\_{\substack{\mathbf{k} \in \{\{\mathbf{l}\}, \{\mathbf{l}\}, \{\text{III}\}\}}} \left(-\Delta H\right)\_{\text{k}} R\_{\text{k}} \tag{28}$$

Table 3. Cold passage dynamic mathematical model.

For the solid structures, such as the fins and the separators, the temperature is considered to be uniform at the same cross-section. The energy conservation equation is written as:

$$\frac{\partial T\_{\rm w}}{\partial t} = K \frac{\partial^2 T\_{\rm w}}{\partial \mathbf{x}^2} + \frac{\alpha\_{\rm h} S\_{\rm h} \eta\_{0, \rm h}}{M\_{\rm w} \mathbb{C} p\_{\rm w}} (T\_{\rm w} - T\_{\rm h}) + \frac{\alpha\_{\rm c} S\_{\rm c} \eta\_{0, \rm c}}{M\_{\rm w} \mathbb{C} p\_{\rm w}} (T\_{\rm w} - T\_{\rm c}) \tag{29}$$

d 2 () () <sup>d</sup> dx *<sup>x</sup>*

In addition to the configuration and geometry parameters of the heat exchange reformer, as shown in Table 5, and fluid properties calculated at the local position, some boundary conditions were also required to carry out the simulation. These included inlet flow rate, fluid composition, and the inlet temperature and outlet pressure of both the hot and cold

*K T T T T*

2 h,2 w,2 c,2 w,2 w w w w

(37)

 

w,2 w,3 w,2 w,1 h,2 h c,2 c

*t M Cp M Cp*

*T T TT S S*

Table 4. Heat exchange reformer volume-resistance characteristic model.

System geometry parameters Length 1 m Width 0.5 m Height 0.532 m

Width 4.5E-3 m Height 6.5E-3 m Offset strip fin length 0.05m Fin thickness 3.0E-3 m

Width 4.5E-3 m Height 5.0E-3 m Offset strip fin length 0.05m Fin thickness 5.0E-3 m

Thickness 1.0E-3 m

Density 3100 kgm-3 Heat capacity 0.640 kJkg-1K-1 Thermal conductivity 0.080 kJm-1s-1K-1

thickness 5.0E-5 m Density 2355 kgm-3 Catalyst reduced activity 0.003

Table 5. Geometry and properties parameters of heat exchange reformer.

Solid structure properties (SiC ceramic [27-29])

Hot passage

Cold passage

Separator

Catalyst properties

Solid structure

**4.2 Simulation conditions** 

streams (Table 6).

The heat conductivity coefficient is *K L A M Cp* ww w w / , the cross area of solid structure is w hh hh h cc cc <sup>c</sup> *A Wt n X Y t t n X Y t t* 2 , and the mass is *M*w ww *A L* .

The control equations of the heat exchange reformer are strongly coupled. In addition to the partial differential equations presented above, two perfect state equations *P=f*(,*T*) for the hot and cold passages are also needed in order to compose a close equation set.

### **4. Simulation modelling and conditions**

#### **4.1 Volume-resistance characteristic model**

In general, nonlinear partial differential equations are treated numerically. However, stability is one crucial factor when using a difference algorithm. In addition, the time step for the difference algorithm is usually very short, so the numerical process is very time consuming [4].

In order to avoid the coupled iteration between the flow rate and pressure, the volumeresistance characteristic modeling technique [4, 22] is introduced into the heat exchange reformer. This modeling technique is based on the lumped-distributed parameter method, which can obtain a set of ordinary differential equations from partial differential equations.

The volume-resistance characteristic model is listed in Table 4 in detail.

Hot passage

$$\frac{\text{d}P\_{\text{h},1}}{\text{d}t} = \frac{RT\_{\text{h},1}}{M\_{\text{h}}A\_{\text{h}}} \frac{G\_{\text{h},1} - G\_{\text{h},2}}{\text{d}\mathbf{x}} \tag{30}$$

$$\frac{\text{dG}\_{\text{h},2}}{\text{d}t} = A\_{\text{h}} \frac{P\_{\text{h},1} - P\_{\text{h},2}}{\text{d}\infty} - \mathcal{U}\_{\text{h}} \sigma\_{\text{h},2} \tag{31}$$

$$\frac{\mathrm{d}T\_{\mathrm{h},2}}{\mathrm{d}t} = -\frac{G\_{\mathrm{h},2}}{A\_{\mathrm{h}}\rho\_{\mathrm{h},2}} \frac{T\_{\mathrm{h},1} - T\_{\mathrm{h},2}}{\mathrm{d}\mathrm{x}} - \frac{S\_{\mathrm{h}}a\_{\mathrm{h},2}}{\rho\_{\mathrm{h},2}Cp\_{\mathrm{h},2}A\_{\mathrm{h}}L} (T\_{\mathrm{h},2} - T\_{\mathrm{w},2}) \tag{32}$$

Cold passage

$$\frac{\text{dC}\_{\text{c},i,2}}{\text{d}t} = -\frac{u\_{\text{c},2}\text{C}\_{\text{c},i,2} - u\_{\text{c},1}\text{C}\_{\text{c},i,1}}{\text{d}\mathbf{x}} + \sum\_{\text{k}\in\{\text{(1)},\text{(1)},\text{(II)}\}} v\_{i,\text{k}}R\_{k,2}\frac{1}{Y\_{\text{c}}} \quad \text{i} \in \{\text{CH}\_{4},\text{H}\_{2},\text{CO},\text{CO}\_{2},\text{H}\_{2}\text{O}\} \tag{33}$$

$$\frac{\text{d}P\_{\text{c,2}}}{\text{d}t} = \frac{\text{RT}\_{\text{c,2}}}{M\_{\text{c}}A\_{\text{c}}} \frac{\text{G}\_{\text{c,2}} - \text{G}\_{\text{c,1}}}{\text{dx}} \tag{34}$$

$$\frac{\text{d}G\_{\text{c},1}}{\text{d}t} = A\_{\text{c}} \frac{P\_{\text{c},2} - P\_{\text{c},1}}{\text{d} \text{x}} - \text{U}\_{\text{c}} \sigma\_{\text{c},1} \tag{35}$$

$$\frac{\text{d}T\_{\text{c1}}}{\text{d}t} = \frac{\text{G}\_{\text{c1}}}{A\_{\text{c}}\rho\_{\text{c1}}} \frac{T\_{\text{c1}} - T\_{\text{c2}}}{\text{d}\mathbf{x}} - \frac{S\_{\text{c}}a\_{\text{c1}}}{\rho\_{\text{c1}}\mathbb{C}\rho\_{\text{c1}}A\_{\text{c}}L} (T\_{\text{c1}} - T\_{\text{w},1}) + \frac{1}{\rho\_{\text{c1}}\mathbb{C}\rho\_{\text{c1}}Y\_{\text{c}}} \sum\_{\text{k}\in\{\{\text{l}\},\{\text{ll}\},\{\text{ll}\}\}} \left(-\Delta H\right)\_{\text{k},1}R\_{\text{k},1} \tag{36}$$

Solid structure

228 Heat Exchangers – Basics Design Applications

The control equations of the heat exchange reformer are strongly coupled. In addition to the

In general, nonlinear partial differential equations are treated numerically. However, stability is one crucial factor when using a difference algorithm. In addition, the time step for the difference algorithm is usually very short, so the numerical process is very time

In order to avoid the coupled iteration between the flow rate and pressure, the volumeresistance characteristic modeling technique [4, 22] is introduced into the heat exchange reformer. This modeling technique is based on the lumped-distributed parameter method, which can obtain a set of ordinary differential equations from partial differential equations.

> h,1 h,1 h,1 h,2 h h

> > *A U*

 

h h h,2

, ,2 I , II , III c

*<sup>i</sup>* CH ,H ,CO,CO ,H O 42 22 (33)

*ik k*

c c c,1

 

(36)

*v R*

c,2 c,2 c,2 c,1 c c

*A U*

d dx *P RT G G*

c,1 c,2 c,1

c c,1 c,1 c,1 c c,1 c,1 c , , <sup>d</sup> <sup>1</sup> ( ) <sup>d</sup> dx *<sup>k</sup> <sup>k</sup>*

(32)

d dx *P RT G G*

h h,2 h,2 h,2 h <sup>d</sup> ( ) d dx

h,2 h,1 h,2

*t MA*

d dx *G PP*

h,2 h,2 h,1 h,2 h h,2

*t A Cp A L*

*t MA*

d dx *G PP*

*T GTT S*

*k*

d

*t*

*t A Cp A L Cp Y* 

ww w w / , the cross area of solid structure is

(30)

(31)

h,2 w,2

(34)

(35)

c,1 w,1 ,1 ,1

*T T H R*

*k I II III*

*T T*

*A L* .

,*T*) for the

w hh hh h cc cc <sup>c</sup> *A Wt n X Y t t n X Y t t* 2 , and the mass is *M*w ww

partial differential equations presented above, two perfect state equations *P=f*(

hot and cold passages are also needed in order to compose a close equation set.

The volume-resistance characteristic model is listed in Table 4 in detail.

d

*t*

d

d 1

*t Y*

d

c, ,2 c,2 c, ,2 c,1 c, ,1

c,1 c,1 c,1 c,2 c c,1

*T GTT S*

*i ii*

*C uC uC*

d dx

The heat conductivity coefficient is *K L A M Cp*

**4. Simulation modelling and conditions 4.1 Volume-resistance characteristic model** 

consuming [4].

Hot passage

Cold passage

$$\frac{\text{d}T\_{\text{w},2}}{\text{d}t} = K\_x \frac{T\_{\text{w},3} - 2T\_{\text{w},2} + T\_{\text{w},1}}{\text{(d}\infty\text{)}^2} - \frac{a\_{\text{h},2}S\_{\text{h}}}{M\_{\text{w}}\text{C}p\_{\text{w}}} (T\_{\text{h},2} - T\_{\text{w},2}) - \frac{a\_{\text{c},2}S\_{\text{c}}}{M\_{\text{w}}\text{C}p\_{\text{w}}} (T\_{\text{c},2} - T\_{\text{w},2}) \tag{37}$$

Table 4. Heat exchange reformer volume-resistance characteristic model.

#### **4.2 Simulation conditions**

In addition to the configuration and geometry parameters of the heat exchange reformer, as shown in Table 5, and fluid properties calculated at the local position, some boundary conditions were also required to carry out the simulation. These included inlet flow rate, fluid composition, and the inlet temperature and outlet pressure of both the hot and cold streams (Table 6).


Table 5. Geometry and properties parameters of heat exchange reformer.

The temperature profiles of the cold stream, hot stream, and solid structure along the heat exchange reformer length are presented in Fig. 5. Because of the high endothermic methane reforming reaction, the cold fuel temperature decreases a little at the entrance. Then, the cold fuel temperature increases along its flow direction due to the heat transfer from hot gas. The temperatures of the hot gas stream and the solid structure decrease along the heat exchange reformer length. It should be noted that the temperature curve is just the line

Fig. 4. Fuel molar fraction along the heat exchange reformer length.

between measured points, so it can't indicate the trend at both ends.

Fig. 5. Temperature distribution along the heat exchange reformer length.


Table 6. Key simulation parameters under the basic condition.

At the same time, some simplifying conditions are used to solve the equations; for example, the heat flux of both the solid structure at inlet and outlet are considered to be zero. As a result, contrasted to the centre difference algorithm in the middle of the solid structure, the difference algorithms for both the front and end modules are treated independently.
