**3. Type-1: two fluids direct contact (TFDC) exchanger**

TFDC heat exchangers, such as air washer chamber, cooling tower, and swamp cooler or direct contact evaporative cooler, are the first type of DCHME shown in **Figure 2**. TFDC has two water and air fluids flowing in parallel or counter, and directly contacting each other to exchange heat and mass between them. **Figure 6(a)** shows the basic one discretized element of air washer spray chamber and direct contact evaporative cooler. The air stream is directly in contact with the saturated layer of each water droplet or the fill/cool pad, and heat and mass are transferring between them according to the mass transfer coefficient *KM* with mass transfer area *aM* and heat transfer coefficient *α<sup>a</sup>* with heat transfer area *aH*. **Figure 6(b)** shows the fundamental discretized element of the cooling tower and swamp cooler. Both elements have the same concept of numbering node for three layers parameters; air and its average, water and its average, and saturated layer parameters, shown in **Figure 6(c)**.

## **3.1 Energy balance between two fluids of Type-1 model (energy balance line, EBL)**

In TFDC exchanger, the water is sprayed into the chamber along the airflow (parallel flow), shown in **Figure 5(a)**, or against the airflow (counter flow), shown in **Figure 5(b)**. Due to the difficulty of area measurement, TFDC exchangers are discretized in length along the fluids flowing, assuming it has same cross sectional area. **Figure 5** shows that the system is divided into nine differential volumes numbered by "*j*," and the inlet and outlet of the differential volumes are noted by "*n*." Since each differential volumes has the same cross sectional area, the differential volume can be changed with differential length "*dL*." and mass flowrates of water and air are needed to be changed to mass flux or flow rate per unit cross-sectional area of the exchanger, *ACS* [m2 ], for air *Ga* [kg/(sm2 )], and water *GL* [kg/(sm2 )].

According to the conservation of energy, the energy transferred from one fluid is equal to the energy gain of the other fluid. Assuming that heat loss from the

#### **Figure 6.**

*(a) Single discretized element of the air washer spray chamber or direct contact evaporative cooler, and (b) of cooling tower or swamp cooler, and (c) node numbering of the points for air stream, water flow, and saturated layer of each element.*

*Direct Contact Heat and Mass Exchanger for Heating, Cooling, Humidification… DOI: http://dx.doi.org/10.5772/intechopen.102353*

system and the change of air or water flow rate due to evaporation or condensation are comparatively small and negligible. Hence, the energy equation of two fluids can be expressed as follows:

$$G\_{a}dh\_{a} = -G\_{l}Cp\_{l}dT\_{l} \tag{7}$$

where, *dha* [kJ/kg] and *dTl* [°C] are the enthalpy and temperature difference between outlet and inlet of one discretized element, and Eq. (7) can be rewritten with element node number for the parallel and counter flow as follows:

$$\frac{h\_a^{n+1} - h\_a^n}{T\_l^{n+1} - T\_l^n} = -\frac{G\_l C p\_l}{G\_a} \tag{8}$$

$$\frac{h\_a^{n+1} - h\_a^n}{T\_l^n - T\_l^{n+1}} = -\frac{G\_l G p\_l}{G\_a} \tag{9}$$

The generalized energy balance line equation with node number for the counter and parallel flow can be described as follows:

$$\frac{h\_a^{n+1} - h\_a^n}{T\_l^{n+1} - T\_l^n} = \pm \frac{G\_l C p\_l}{G\_a} \tag{10}$$

A minus sign refers to parallel flow of air and water stream. A plus sign refers to counter flow (water flow is in the opposite direction of airflow).

#### **3.2 Heat and mass transfer between two fluids of Type-1 model (tie-line slope, TLS)**

According to the conservation of mass between the two fluids, change of water amount in liquid stream, *dGl*, due to evaporation (if saturated humidity ratio, *ws* [kg/kg], at the saturated layer is greater than humidity ratio, *wa*, of air stream) or condensation (if *ws* < *wa*) is equal to the amount of humidification or dehumidification of air stream, �*Gadw*, in the spray chamber. The above statement can be expressed by equation as follows:

$$
\Delta \text{mass} = dG\_l = -G\_a dw\_a \tag{11}
$$

The negative sign means that the increasing of mass in one fluid stream is equal to the losing mass of other fluid stream. Similarly, the change of mass flow rate, Δ*mass* from either stream is equal to the mass flux [kg/m<sup>2</sup> ], that is, mass transfer from the air stream to the saturated liquid surface (or) vise visa per contact surface area along the exchanger, *ACT* [m<sup>2</sup> ]. It can be written in equation as follows:

$$
\Delta m \text{as} = K\_M a\_M (w\_s - w\_{aavg}) dL \tag{12}
$$

where, *KM* [kg/(s�m2 )] is mass transfer coefficient, that is, mass transfer rate per contact surface area along the exchanger, *ACT* [m2 ]. *aM* [m<sup>2</sup> /m<sup>3</sup> ] is a mass transferring contact surface area between air and water per unit volume of spray chamber. *L* [m] is the chamber length. The sensible heat transfer between air stream and saturated liquid surface can be expressed as follows:

$$-G\_a C\_{p\text{m}} dT\_a = a\_a a\_H (T\_s - T\_{a\text{avg}}) dL \tag{13}$$

where, *α<sup>a</sup>* [W/(m<sup>2</sup> �K)] is convective heat transfer coefficient of air. *aH* [m2 /m<sup>3</sup> ] is heat transfer contact surface area between air and water per unit volume of spray chamber. *Cpm* [kJ/(kgda�K)] is a specific heat of moist air at constant pressure. *Ts* [°C] is saturation temperature of water film. *Taavg* [°C] is the average air temperature of two adjacent nodes, see in **Figure 6(c)**. The total heat transfer between air stream and saturated liquid surface is equal to the sum of sensible heat, *CpmdTa*, and latent heat, *hfgdw*. The total heat transfer can be calculated by combining of mass transfer Eq. (12) multiplying with enthalpy of vaporization *hfg* [kJ/kg] and the sensible heat transfer Eq. (13). The total heat transfer equation can be expressed as follows:

$$\mathbf{G}\_{a}\left(\mathbf{C}p\_{m}dT\_{a} + h\_{\text{fg}}dw\right) = \left[\mathbf{K}\_{M}a\_{M}\left(w\_{s} - w\_{\text{avg}}\right)h\_{\text{fg}} + a\_{a}a\_{H}\left(T\_{s} - T\_{a\text{avg}}\right)\right]d\mathbf{L} \tag{14}$$

Since *CpmdTa* þ *hfgdw* is equal with enthalpy change of air *dha*, and a small change of water vaporization heat with the function of temperature is neglected, the Eq. (14) can be rewritten as follows:

$$\mathbf{G}\_{a}d\mathbf{h}\_{a} = \mathbf{K}\_{M}\mathbf{a}\_{M} \left[ (\boldsymbol{w}\_{\text{s}} - \boldsymbol{w}\_{\text{avg}}) \boldsymbol{h}\_{\text{fg}} + \frac{\mathbf{a}\_{a}\boldsymbol{a}\_{H}}{\mathbf{K}\_{M}\mathbf{a}\_{M}} \left( T\_{\text{s}} - T\_{a\text{avg}} \right) \right] d\boldsymbol{L} \tag{15}$$

The contact surface areas per system volume, *aH* and *aM* for the heat and mass transferring process in spray chambers are identical (*aH* = *aM*), however, this hypothetical is not always correct, especially, if the systems are using the packing materials as an extension of contact surface area which are not fully wetted. Assuming Lewis number with the power of 2/3 is similar with 1, see in Eq. (6), and neglecting of a small variations in *hfg*. McElgin and Wiley [25] simplified Eq. (15) as follows:

$$\mathbf{G}\_{\rm d}dh\_{\rm at} = \mathbf{K}\_{\rm M}\mathbf{a}\_{\rm M} \left( h\_{\rm s} - h\_{\rm aavg} \right) dL \tag{16}$$

The sensible heat transferring from saturated liquid surface to the working fluid stream is equal to the heat gain of the working fluid, the energy balance equations can be depicted as follows:

$$\mathbf{G}\_l \mathbf{C} p\_l d\mathbf{T}\_l = a\_l a\_H (T\_s - T\_{\text{law}}) d\mathbf{L} \tag{17}$$

After substitution of Eqs. (16) and (17) into Eq. (7), and rearranged, the Eq. (7) can be written as follows:

$$-a\_l a\_H \left( T\_s - T\_{lavg} \right) dL = K\_M a\_M \left( h\_s - h\_{aavg} \right) dL \tag{18}$$

$$\frac{h\_t - h\_{\text{aug}}}{T\_t - T\_{\text{long}}} = -\frac{a\_l a\_H}{K\_M a\_M} = -\frac{a\_l \text{ C} p\_m}{a\_a} \tag{19}$$

According to the above Eq. (19), the value of, � *<sup>α</sup><sup>l</sup> Cpm <sup>α</sup><sup>a</sup>* , is a slope or gradient line of the rise, the enthalpy incensement in vertical over the run, temperature incensement along the horizontal in T-h graph. In other words, the slope is a ratio of the enthalpy difference between air and saturated surface to the temperature difference between working fluid and saturated surface. This slope, the relation of the temperature and enthalpy, can be denoted as tie-line slope,*TLS* of the Type-1 TFDC exchanger. The *TLS*, Eq. (19), can be solved by using graphical method, drawing in water-air T-h graph which is proposed in [2, 19, 20, 26], but this method cannot be applied in the numerical model. In order to solve TLS equation numerically, the two unknown parameters, saturated temperature,*Ts*, and saturated enthalpy, *hs*, have to *Direct Contact Heat and Mass Exchanger for Heating, Cooling, Humidification… DOI: http://dx.doi.org/10.5772/intechopen.102353*

reduce one unknown parameter by substitution of their relation equation that can be obtained by fitting of the saturation line on T-h graph, calculated from the psychometric chart equations depicted in ASHRAE code [19], with the third order polynomial regression equation, Eq. (20).

$$h\_t = \text{coef1}.T\_s^3 + \text{coef2}.T\_s^2 + \text{coef3}.T\_s + \text{coef4} \tag{20}$$

**Figure 7** shows the third order polynomial regression line with the saturation point, in red circle legend, calculated by psychometric chart equations, and its four related coefficients values, *coe*1 to *coe*4. The regression has a very high accuracy as its R-square value 0.9993706, which covering the range of air temperature from 0 to 50°C.

By the substitution of polynomial regression equation Eq. (20) into Eq. (19), the tie-line slope equation become one unknown parameter equation, and that can be rewritten with node number *n* and *j* as follows:

$$\begin{aligned} \left(co\mathbf{e1}\left(T\_s^{\dagger}\right)^3 + co\mathbf{e2}\left(T\_s^{\dagger}\right)^2 + (co\mathbf{e3} + T\mathbf{L}S)T\_s^{\dagger} + co\mathbf{e4} - \left(\frac{h\_a^n + h\_a^{n+1}}{2}\right) - T\mathbf{L}S\left(\frac{T\_l^n + T\_l^{n+1}}{2}\right)\right) \\ = \mathbf{0} \end{aligned} \tag{21}$$

Newton Raphson iteration method or any other relevant numerical method can be applied to solve out the unknown root value *Ts* of each node *j* of the equation, Eq. (21), by using the known value of *Tl* and *ha* of each node *n* (node numbering of each element is shown in **Figure 6**).

#### **3.3 Dry bulb temperature of each node for Type-1 model**

The last unknown parameter of air temperature,*Ta* [°C], of each node n, can be calculated from the ratio of sensible heat Eq. (13) to total heat and mass transfer Eq. (16), and it gives as follows:

$$\frac{dT\_a}{dh\_a} = \frac{a\_a a\_H \left(T\_s - T\_{a \text{avg}}\right)}{K\_M C\_{pm} a\_M \left(h\_s - h\_{a \text{avg}}\right)}\tag{22}$$

**Figure 7.** *Third order polynomial regression equation of the enthalpy of the saturated air.*

The value of *<sup>α</sup><sup>a</sup> KMCpm* can be taken as 1 because the Lewis number with the power of 2/3 is approximately equal to 1, see in Eq. (6). For the TFDC exchanger, this study assumes that the contact surface area per volume of heat transferring process, *aH*, is identical with the one of mass transferring process, *aM*. For numerical calculation, the average parameters,*Taavg* and *haavg*, are located between two adjacent n nodes, and noted with *j* node, shown in **Figure 6**. After rearranging of the Eq. (22), it gives with node number *n* and *j* as follows:

$$T\_a^{n+1} - T\_a^n = \frac{\frac{h\_a^{n+1} - h\_a^n}{\left(h\_i^j - h\_{\text{avg}}^j\right)} \left(T\_s^j - T\_a^n\right)}{\left(\mathbf{1} + \frac{h\_a^{n+1} - h\_a^n}{2\left(h\_i^j - h\_{\text{avg}}^j\right)}\right)}\tag{23}$$

The outlet air temperature of each node, *Tn*þ<sup>1</sup> *<sup>a</sup>* , can be calculated from the known inlet temperature, *T<sup>n</sup> <sup>a</sup>*, air enthalpy, *ha*, average air enthalpy, *haavg* , vapor saturated air enthalpy, *hs*, and vapor saturated air temperature, *Ts* by using Eq. (23).

### **3.4 Contact length calculation for Type-1 TFDEC exchanger model**

Calculation of the contact length or depth of packing of the exchanger, *L* [m], is a primary interest for the designing of the system to achieve a desired out let condition of air or out let water temperature. The length can be calculated by the integration of heat and mass transfer equation, Eq. (16).

$$L = \frac{G\_a}{K\_M a\_M} \int\_{j=1}^{j=N-1} \frac{1}{\left(h\_s - h\_{aavg}\right)} dh\_a \tag{24}$$

The integral equation, Eq. (24) can be solved by using Sampson's rule or Trapezoidal integration method. The integral equation can be rearranged and rewritten with node number *n* and *j* as follows:

$$L = \frac{G\_a}{K\_{M\mathfrak{A}\_M}} \sum\_{j=1}^{j=N-1} \left[ \frac{\mathbf{1}}{\left(h\_s^j - h\_{a\text{avg}}^j\right)} \times \left(h\_a^{n+1} - h\_a^n\right) \right] \tag{25}$$

Change of the air mass flow rate, *Ga*, due to the transfer of water vapor from the water droplet or film to the air by evaporation or condensation is very minimal compared with its mass flowrate, and it can be taken as negligible for all the practical applications. The contact length or exchanger length *L*, can be calculated from the known air enthalpy, *ha*, and saturated enthalpy, *hs*, of each elements.
