2. Fundamentals

#### 2.1. Steady-state operation

Heat transfer in agitated vessels is carried through heat exchange surfaces, like jackets, helical coils, spiral coils, and vertical tubular baffles [5]. The surfaces of heat exchange are designed as a function of the area necessary to carry the heating or cooling, based on the overall heat transfer coefficient, which is a function of the dimensionless groups Reynolds, Prandtl, and Nusselt. The classic project equation of the heat exchange area for steady-state operation is shown in Eq. (1).

$$A = \mathbb{Q}/\text{ULMTD} \tag{1}$$

in which the overall heat transfer coefficient (U) is a function of local coefficients of internal convection, related to the external surface area or internal and external convection coefficient (hi and ho) for the hot and cold fluid, respectively.

The jackets are surfaces of heat transfer characterized by encompassing the tank, given that the thermal fluid roams the space between the jacket and the tank. On these equipments, the heat transfer efficiency is low due to the heat source being in the wall, which provokes uneven heating of the fluid within the tank, besides the structural problems in big units. Nevertheless, this system presents advantages as easy cleaning and available project data provided in the

The helical coils are made out of tubes between the impeller center line and the tank's wall. The contact area with the fluid to be heated or cooled is much bigger around the jackets, which end up increasing the heat transfer efficiency; however, the difficulty is in the cleaning of the equipment. Most agitated vessels involving heat transfer operations are projected with helical

The spiral coils consist of wounded tubes generally placed on the bottom of the tank. They are mostly applied to heating viscous fluids in pumping transport. Spiral coils have as disadvantage the located heat transfer, fostering uneven heating. Another downside is the lack of data

The three heating transfer surfaces mentioned above demand baffles alongside the tank's wall to avoid vortex, which is characterized by the development of a bottleneck formed by the agitated fluid around the mechanical impeller. The vortex is an indicator of inefficiency at mixing and low heat transfer, due to the circular and organized stream

The vertical tube baffles are surfaces that provide significant heating transfer rate and also eradicate the development of a bottleneck, due to its spatial structure, which makes necessary to allocate extra baffles alongside the tank's wall. The drawback of this technique is the lack of

However, each type of heat transfer surface will present parameters for Nusselt's equation particular to each system in which they were determined, in such a way that it is possible to

For batch processes the heat transfer area is determined by transient energy balances, while for continuous processes, on the large majority, the area is obtained by energy balances for steady-

Heat transfer in agitated vessels is carried through heat exchange surfaces, like jackets, helical coils, spiral coils, and vertical tubular baffles [5]. The surfaces of heat exchange are designed as a function of the area necessary to carry the heating or cooling, based on the overall heat transfer coefficient, which is a function of the dimensionless groups Reynolds, Prandtl, and Nusselt. The classic project equation of the heat exchange area for steady-state operation is

obtain a general model that covers all surfaces for heat transfer in agitated vessels.

current literature [1].

in the current literature [3].

lines.

state operation.

2. Fundamentals

shown in Eq. (1).

2.1. Steady-state operation

coils due to the availability of project data [2].

38 Heat Exchangers– Design, Experiment and Simulation

data project, especially on continuous operation [4].

Eq. (2) shows the dependence of the overall coefficient U to the convection coefficients (internal and external), the resistance to heat transfer presented by the heat exchange surface (conduction), and the resistances by internal and external fouling in relation to the pipelines.

$$1\,\text{l/L}A = 1/h\_iA\_i + Rd\_i/A\_i + R\_c + Rd\_0/A\_0 + 1/h\_oA\_o\tag{2}$$

The thermal resistance presented by conduction can be disregarded when the thickness of the surface wall is negligible in relation to the internal and external diameter and when it is made out of a material with high thermal conductivity. Therefore, the term Rc (referring to the thermal resistance presented by heat conduction) in Eq. (2) can be disregarded [6]. The terms regarding the resistance by fouling can also be neglected if the pipelines used are new or with operational time inferior to 5 years. Eq. (2) can be written as shown in Eq. (3), being the overall heat transfer coefficient only dependent of the internal and external convection.

$$1/\mathsf{UIA} = \mathbbm{1}/h\_i A\_i + \mathbbm{1}/h\_o A\_0 \tag{3}$$

The internal film coefficient by definition is associated to the fluid's flow in the interior of the heat exchange surface, while the external coefficient, in the case of a vessel's surface, is related to the mixture of the fluids that will be agitated. If the mixture, which will be agitated, is the one that will be heated, the heat transfer happens on the external surface of contact of the hot source with the cold fluid; therefore, the internal convection coefficient must be corrected and placed as a function of this external area, called then as corrected internal coefficient (hio), calculated from the relation between the internal and external diameter of the pipeline, as shown in Eq. (4).

$$h\_{i0} = h\_i \frac{D\_i}{D\_e} \tag{4}$$

Adding Eq. (4) to Eq. (3) and indicating the areas as a function of the diameters, Eq. (5) is obtained.

$$1/\mathcal{U}\pi \mathcal{D}\_{\mathfrak{c}}L = 1/(\mathcal{D}\_{\mathfrak{c}}/\mathcal{D}\_{\mathfrak{i}})h\_{\mathfrak{i}\mathfrak{o}}\pi \mathcal{D}\_{\mathfrak{i}}L + 1/h\_{\mathfrak{o}}\pi \mathcal{D}\_{\mathfrak{c}}L\tag{5}$$

It can be seen in Eq. (5) that the area of heat transfer A (πDeL) corresponds to the external area of the tube's surface Ae. Grouping the common terms in Eq. (5), the overall coefficient of heat transfer U can be calculated as a function of the corrected internal convection coefficient and the external convection coefficient, as shown in Eq. (6).

$$\mathbf{1}/\mathsf{U} = \mathbf{1}/\mathsf{h}\_{\mathrm{i}o} + \mathbf{1}/\mathsf{h}\_{o} \tag{6}$$

The determination of the rectified internal convection coefficient (hio) implies the determination of the coefficient hi, which is a function of the fluid's physical properties (viscosity, density, specific heat, and thermal conductivity), the heat transfer surface geometry (tube diameter), the kind of flow (laminar or turbulent), the speed flow, and the temperature gradient [7].

The analytic determination of the equation, which relates the internal film coefficient to the aforementioned variables, can be done through the application of the hydrodynamic and thermal boundary layer theory on the heat transfer surface exposed to the hot fluid and the cold fluid. After using dimension analysis and some similar equations to those of continuity, momentum, and energy, it can be noticed that the Nusselt's number is a function of the Reynolds and Prandtl's number. However, this analytical method can be difficult to be solved due to the nonhomogeneous partial differential equations, which will depend of the specific boundary conditions as a function of the heat transfer surface geometry. To avoid this situation, the Buckingham's Pi theorem can be used, which relates the variables that may affect the answer variables, in this case the film coefficient, through dimensionless groups [8]. With dimension analysis the internal film coefficient can be related with the many variables aforementioned with a function in which hi is the dependent variable, as seen in Eq. (7).

$$\mathcal{H}\_i = f\left(\mu, \rho, c\_p, D\_i, k, u\right) \tag{7}$$

The function expressed in Eq. (7) can be written powering the dependent variables to an exponent, adding a proportionality constant K and a dimensional constant Kh, which will be reduced to 1,0 if all the other variable dimensions, when combined, lead to thermal quantities, as illustrated in Eq. (8).

$$\mathbf{M}\_i = \mathbf{K} \boldsymbol{\mu}^a \boldsymbol{\rho}^b \mathbf{c}\_p^c \,^c \mathbf{D}\_i^d \, \mathbf{k}^e \boldsymbol{\mu}^f \mathbf{K}^{\prime g} \tag{8}$$

Expressing all the terms in Eq. (8) as fundamental quantities and then rearranging it, the semiempiric model known as Nusselt's equation can be obtained (Eq. (9)).

$$\frac{h\_i.D\_i}{k} = K \left(\frac{D\_i u \rho}{\mu}\right)^a \left(\frac{c\_p \mu}{k}\right)^b \tag{9}$$

The term on the left side of Eq. (9) is the dimensionless Nusselt, and the terms on the right, going from left to right, are the dimensionless Reynolds and Prandtl, respectively; hence, Eq. (9) can be written in the following manner.

$$N\mu = \text{K.Re}^a.Pr^b \tag{10}$$

Many researches established the exponents and constant values in Eq. (24) for forced flow on the interior of the cylindrical heat transfer surface, that is, on the interior of the pipelines.

Sieder-Tate [9] added a correction factor in Eq. (9), taking into account the viscosity effects on wider variations of temperature, a relation between the fluid's viscosity on the bulk temperature, and the fluid's viscosity on the wall's temperature, as shown in Eq. (11), which is valid for turbulent flows on the interior of cylindrical surfaces with Reynolds above 10,000 and smooth pipes.

Design of Heat Transfer Surfaces in Agitated Vessels http://dx.doi.org/10.5772/66729 41

$$Nu = 0.027 Re^{0.8} Pr^{1/3} \left(\mu/\mu\_w\right)^{0.14} \tag{11}$$

Eq. (11) is a simplified form to determine the internal coefficient film; however, it carries an overall error of 40%, being necessary to implement correction factors on the design of the heat transfer area. Geankoplis [10] presents an equation to determine the internal coefficient film for water with temperatures ranging from 4 to 105°C on smooth pipes' interior in fully developed turbulent flow, as shown in Eq. (12), which carries an overall error of 25%.

specific heat, and thermal conductivity), the heat transfer surface geometry (tube diameter), the kind of flow (laminar or turbulent), the speed flow, and the temperature gradient [7].

The analytic determination of the equation, which relates the internal film coefficient to the aforementioned variables, can be done through the application of the hydrodynamic and thermal boundary layer theory on the heat transfer surface exposed to the hot fluid and the cold fluid. After using dimension analysis and some similar equations to those of continuity, momentum, and energy, it can be noticed that the Nusselt's number is a function of the Reynolds and Prandtl's number. However, this analytical method can be difficult to be solved due to the nonhomogeneous partial differential equations, which will depend of the specific boundary conditions as a function of the heat transfer surface geometry. To avoid this situation, the Buckingham's Pi theorem can be used, which relates the variables that may affect the answer variables, in this case the film coefficient, through dimensionless groups [8]. With dimension analysis the internal film coefficient can be related with the many variables aforementioned with a function in which hi is the dependent variable, as

The function expressed in Eq. (7) can be written powering the dependent variables to an exponent, adding a proportionality constant K and a dimensional constant Kh, which will be reduced to 1,0 if all the other variable dimensions, when combined, lead to thermal quantities,

hi ¼ Kua

semiempiric model known as Nusselt's equation can be obtained (Eq. (9)).

hi:Di

Eq. (9) can be written in the following manner.

ρb cp c Di d k e μf

k ¼ K Diuρ

Expressing all the terms in Eq. (8) as fundamental quantities and then rearranging it, the

μ a cpμ

The term on the left side of Eq. (9) is the dimensionless Nusselt, and the terms on the right, going from left to right, are the dimensionless Reynolds and Prandtl, respectively; hence,

Nu ¼ K:Rea

Many researches established the exponents and constant values in Eq. (24) for forced flow on the interior of the cylindrical heat transfer surface, that is, on the interior of the pipelines.

Sieder-Tate [9] added a correction factor in Eq. (9), taking into account the viscosity effects on wider variations of temperature, a relation between the fluid's viscosity on the bulk temperature, and the fluid's viscosity on the wall's temperature, as shown in Eq. (11), which is valid for turbulent flows on the interior of cylindrical surfaces with Reynolds above 10,000 and

k b

hi ¼ f μ; ρ; cp; Di; k; u (7)

K′′g (8)

:Prb (10)

(9)

seen in Eq. (7).

40 Heat Exchangers– Design, Experiment and Simulation

as illustrated in Eq. (8).

smooth pipes.

$$h\_i = 1429 \left(1 + 0.0146 \overline{T}\right) .u^{0.8} / D\_i^{0.2} \tag{12}$$

Eq. (11) and Eq. (12) are simple to be used and present satisfactory results for the majority of heat exchanger projects, despite the high error associated with them. Gnielinski [11] presents a correlation to determine the internal convection coefficient, with an error lower than 10%, for fully developed turbulent flows, with Reynolds ranging from 3000 to 5 · 106 , Prandtl between 0.5 and 2000, and for a relation (L/Di) greater than 10. The correlation is presented in Eq. (13).

$$Nu = \frac{(f'/8)(Re - 1000)Pr}{1 + 12.7(f'/8)^{1/2}\left(Pr^{2/3} - 1\right)}\tag{13}$$

The friction factor (f′) can be obtained from Moody's diagram or by specialized correlations. In the case of Eq. (13), which is for fully developed turbulent flow, the friction factor can be calculated through Eq. (14).

$$f' = \left(0.790lnRe - 1.64\right)^{-2} \tag{14}$$

The external convection coefficient (ho) is a function of the same variables of the internal convection coefficient (hi); however, the heat transfer surface's geometry and the mechanical impeller have large influence in heat transmission. The expression to determine the ho can be obtained through the Buckingham's Pi theorem [5], similar to the determination of hi; hence, Eq. (9) can be rewritten by modifying the Nusselt and Reynolds' numbers for agitation (Rea), presented in Eq. (15), and adding the viscosity correction factor proposed by Sieder-Tate, as shown in Eq. (16).

$$Re\_d = \frac{\text{ND}\_a^{\text{-}2} \rho}{\mu} \tag{15}$$

$$\frac{h\_o D\_t}{k} = K \left(\frac{\text{ND}\_a^{\text{-}} \rho}{\mu}\right)^a \left(\frac{c\_p \mu}{k}\right)^b \left(\frac{\mu}{\mu\_w}\right)^c \tag{16}$$

Differently from Eq. (9), which is generic for any type of fluid, as long as the heat transfer surface is cylindrical and has a fully developed turbulent flow, Eq. (16) has the following limitations: (i) the position and geometry of the mechanical impeller, (ii) the flow's regime, (iii) the geometry of the heat transfer surface, and (iv) the tank's geometry. These limitations do not allow to obtain the values for the K constant and the exponents generically, for all surfaces of heat transfer in agitated vessels [12].

#### 2.2. Unsteady-state operation

The project of the heat transfer surface for agitated vessels will depend on whether the tank is operating in discontinuous mode (batch) or continuous mode.

The batch processes in agitated vessels are common in many industrial processes, specially, in chemical reactors. In batch operations, there is the occurrence of two typical situations: (1) the design parameter is the operation time for heating or cooling; hence, the surface area is unknown; (2) the area for heat exchange is known and the operation time is unknown.

These processes can be carried out on isothermal operations and with phase change (heating with steam or cooling with thermal fluid vaporization) or non-isothermal operations and fluids without phase change.

An important consideration must be highlighted regarding the U coefficient: for heat exchangers in systems operating on countercurrent or cocurrent flow, if the fluid's physical properties do not change significantly during the process, the U coefficient is almost constant over time. However, when the system has at least one of its fluids with considerable variation of physical properties, like viscosity in the case of oils heating, the U coefficient undergoes variation as function of time, making its determination complex, since it is necessary to know the pattern in which it changes over time. Similarly, this consideration can be applied to tanks' heating or cooling.

Considering a vessel with a generic heat transfer surface for heating a fluid using a isothermal source, the heat exchange area is determined from the macroscopic energy balance (Eq. (17)) with the following hypothesis:

$$
\Delta \dot{\boldsymbol{Q}} + \dot{\boldsymbol{w}}\_{\epsilon} \left( \boldsymbol{h}\_{\epsilon} + \frac{\boldsymbol{v}\_{\epsilon}^{2}}{2} + \mathbf{g} \boldsymbol{z}\_{\epsilon} \right) = \frac{d \mathcal{E}\_{\rm rc}}{d \theta} + \dot{\boldsymbol{w}}\_{s} \left( \boldsymbol{h}\_{s} + \frac{\boldsymbol{v}\_{s}^{2}}{2} + \mathbf{g} \boldsymbol{z}\_{s} \right) + \boldsymbol{\Sigma} \dot{\boldsymbol{W}} \tag{17}
$$

Eq. (18) presents a simplification of Eq. (17) following the aforementioned hypothesis.

not allow to obtain the values for the K constant and the exponents generically, for all surfaces

The project of the heat transfer surface for agitated vessels will depend on whether the tank is

The batch processes in agitated vessels are common in many industrial processes, specially, in chemical reactors. In batch operations, there is the occurrence of two typical situations: (1) the design parameter is the operation time for heating or cooling; hence, the surface area is unknown; (2) the area for heat exchange is known and the operation

These processes can be carried out on isothermal operations and with phase change (heating with steam or cooling with thermal fluid vaporization) or non-isothermal operations and

An important consideration must be highlighted regarding the U coefficient: for heat exchangers in systems operating on countercurrent or cocurrent flow, if the fluid's physical properties do not change significantly during the process, the U coefficient is almost constant over time. However, when the system has at least one of its fluids with considerable variation of physical properties, like viscosity in the case of oils heating, the U coefficient undergoes variation as function of time, making its determination complex, since it is necessary to know the pattern in which it changes over time. Similarly, this consideration can be applied to tanks'

Considering a vessel with a generic heat transfer surface for heating a fluid using a isothermal source, the heat exchange area is determined from the macroscopic energy balance (Eq. (17))

¼ dEvc

a. Few physical properties vary in the fluid to be heated; hence, the U coefficient is assumed

c. Perfect mixing tank (in such a way that the temperature will be the same at any point of

e. Workflow in control volume is neglected, being only considered to be significant when the

b. Hot fluid constant flow (on this isothermal heating, water steam will be considered).

dθ þ w\_ s hs þ

vs 2 2 þ gzs

þ ∑W\_ (17)

of heat transfer in agitated vessels [12].

42 Heat Exchangers– Design, Experiment and Simulation

operating in discontinuous mode (batch) or continuous mode.

2.2. Unsteady-state operation

time is unknown.

heating or cooling.

the tank).

d. Perfect tank insulation.

fluid has a large viscosity.

with the following hypothesis:

∑Q\_ þ w\_ e he þ

ve 2 2 þ gze

to be constant over all the heat exchange surface.

fluids without phase change.

$$\dot{w}(h\_{\varepsilon} - h\_{s}) = UA(T - t\_{b}) = Mcp\_{c}\frac{dt\_{b}}{d\varTheta} \tag{18}$$

Integrating Eq. (18) with the specific boundary condition in θ = 0 with tb = tb1 (fluid initial temperature in the beginning of the heating process), Eq. (19) is obtained, which allows to calculate the heat exchange area with a constant U coefficient over the entire surface.

$$A = \frac{Mcp\_c}{L\theta} \ln \frac{T - t\_{b1}}{T - tb\_2} \tag{19}$$

In a similar way, Eq. (20) presents the expression to calculate heat transfer area in the case of isothermal cooling.

$$A = \frac{Mcp\_c}{lI\theta} . ln \frac{t\_{b1} - T'}{t\_{b2} - T'} \tag{20}$$

The energy balance described in Eq. (18) can be applied to nonisothermal processes, where Eq. (21) presents the heat exchange area calculation for heating and Eq. (22) presents the heat exchange area calculation for cooling, highlighting that both equations are valid for hot and cold fluid constant flows and constant inlet temperature.

$$A = \frac{Mcp\_c}{lI} \ln\left\{\frac{1}{1 - \left[\ln\left(\frac{T - t\_{b1}}{T - tb\_2}\right) \frac{Mcp\_c}{\dot{w}\_{hp\_h}\theta}\right]}\right\} \tag{21}$$

$$A = \frac{Mcp\_h}{\mathcal{U}} \ln\left\{ \frac{1}{1 - \left[ \ln\left(\frac{b\_1 - T}{b\_2 - T}\right) \frac{Mcp\_h}{\dot{w}\_{cp\_h \cdot \theta}} \right]} \right\} \tag{22}$$

On continuous operations, the same hypotheses assumed for the processes in batch mode can be considered, adding only that the flow of both fluids will be constant during the whole operation with constant inlet temperatures.

Eqs. (23)–(25) show the obtainment of the heat transfer area for an isothermal process operating at continuous mode.

$$\ln \frac{K\_1 - K\_2 t\_{b2}}{K\_1 - K\_2 t b\_1} = -\frac{K\_2}{M c p\_c} \theta \tag{23}$$

$$K\_1 = \dot{w}\_c c p\_c t\_{b1} + \mathcal{U}AT \tag{24}$$

$$K\_2 = \dot{w}\_c c p\_c + \text{lIA} \tag{25}$$

Similarly, Eqs. (26)−(28) present the heat exchange area for a nonisothermal process operated in continuous mode.

$$
\ln \frac{K\_3 - K\_4 t\_{b2}}{K\_3 - K\_4 t\_{b1}} = -\frac{K\_4}{M c p\_c} \,\Theta \tag{26}
$$

$$K\_3 = \dot{w}\_c c p\_c t\_{b1} + \dot{w}\_h c p\_h \left[ \left( 1 + \frac{1}{e^{\frac{UA}{\dot{w}\_h p\_h}}} \right) + T\_1 \right] \tag{27}$$

$$K\_4 = \dot{w}\_c cp\_c + w\_h cp\_h \tag{28}$$

The equations for continuous operation described in Eqs. (26)–(28) can only be used for processes operated in an unsteady state; however, for projects on tanks operating continuously, they are carried as steady state, in such a way that the design equation described by Eq. (26) is reduced to Eq. (1), as shown in item 1.

### 3. Expressions for Nusselt's number in several heat transfer surfaces

#### 3.1. Jackets

Jackets are surfaces for heat exchange often used by the majority of processes of heating and cooling. Structurally, they consist of an external coating for the tank like a shirt, where the thermal fluid flows through the space between the jacket and the tank. The jackets are mainly classified as standards, spiral, and half-jackets, in which the difference is only the structure; however, these three types have the same placement inside the tank.

The heat transference through these thermal surfaces is not efficient due to its location, since the heat transfer occurs on the tank's walls; therefore, baffles are placed in order to increase the turbulence aiming to implement the heat transfer efficiency [13]. A big disadvantage of using jackets, more than just the low heat transference, is its dimension, since on tanks with large volumes, like fermentation vats with an average volume of 5000 m3 , the construction becomes structurally unfeasible due to the high cost. Industrially, jackets are used in large scale, even with the aforementioned disadvantages, because the exponents and the constant in Eq. (16) are already known for many tank's dimensions and several kinds of fluids operating in continuous or discontinuous mode.

Chilton et al. [14] carried out a pioneer work concerning the determination of the external convection coefficient, in a tank with 0.3 m of diameter for water and glycerin solution heating with axial mechanical impeller. Eq. (29) presents Nusselt's equation with constant and exponents determined by Chilton.

$$Nu = 0.36 Re\_{\mu} ^{0.67} Pr^{0.33} \left(\frac{\mu}{\mu\_w}\right)^{0.14} \tag{29}$$

The exponents and the constant in Eq. (29) were determined in a tank without baffles and discontinuous operation, although, as the dimension of the studied tank was quite inferior to industrial dimensions, Eq. (29) has deviations of 40% magnitude from the real values of external convection coefficients obtained experimentally in tanks of other sizes.

Uhl and Gray [15] carried experiments in a tank with 0.6 m of diameter and axial impeller, turbine type, and anchor type, with ultra-viscous fluids and four vertical baffles, being operated on discontinuous mode. However, because of the elevated viscosity, the equations are only valid if Reynolds ranges between 20 and 300. Eq. (30) presents the expression for the Nusselt determination using the axial impeller.

ln K3 � K4tb2 K3 � K4tb1

K3 ¼ w\_ ccpctb1 þ w\_ hcph 1 þ

Eq. (26) is reduced to Eq. (1), as shown in item 1.

44 Heat Exchangers– Design, Experiment and Simulation

3.1. Jackets

or discontinuous mode.

nents determined by Chilton.

The equations for continuous operation described in Eqs. (26)–(28) can only be used for processes operated in an unsteady state; however, for projects on tanks operating continuously, they are carried as steady state, in such a way that the design equation described by

Jackets are surfaces for heat exchange often used by the majority of processes of heating and cooling. Structurally, they consist of an external coating for the tank like a shirt, where the thermal fluid flows through the space between the jacket and the tank. The jackets are mainly classified as standards, spiral, and half-jackets, in which the difference is only the structure;

The heat transference through these thermal surfaces is not efficient due to its location, since the heat transfer occurs on the tank's walls; therefore, baffles are placed in order to increase the turbulence aiming to implement the heat transfer efficiency [13]. A big disadvantage of using jackets, more than just the low heat transference, is its dimension, since on tanks with large

structurally unfeasible due to the high cost. Industrially, jackets are used in large scale, even with the aforementioned disadvantages, because the exponents and the constant in Eq. (16) are already known for many tank's dimensions and several kinds of fluids operating in continuous

Chilton et al. [14] carried out a pioneer work concerning the determination of the external convection coefficient, in a tank with 0.3 m of diameter for water and glycerin solution heating with axial mechanical impeller. Eq. (29) presents Nusselt's equation with constant and expo-

The exponents and the constant in Eq. (29) were determined in a tank without baffles and discontinuous operation, although, as the dimension of the studied tank was quite inferior to industrial dimensions, Eq. (29) has deviations of 40% magnitude from the real values of

0:67Pr0:33 μ

μw � �0:14

3. Expressions for Nusselt's number in several heat transfer surfaces

however, these three types have the same placement inside the tank.

volumes, like fermentation vats with an average volume of 5000 m3

Nu ¼ 0:36Rea

external convection coefficients obtained experimentally in tanks of other sizes.

¼ � K4 Mcpc

> 1 e U:A w\_ hcph

" #

þ T1

K4 ¼ w\_ ccpc þ whcph (28)

!

θ (26)

, the construction becomes

(29)

(27)

$$Nu = 0.415 Re\_d^{0.67} Pr^{0.33} \left(\frac{\mu}{\mu\_w}\right)^{0.24} \tag{30}$$

Bourne et al. [16] determined the exponents and the constant for Eq. (16) in a tank with diameter of 0.51 m on the standard conditions for dimensions, described on Rushton et al. [17] pioneer work, conditions that are being used up until the present day. Electrolytic solutions and a turbine radial impeller with six flat blades were used; however, Bourne et al. [16] disregarded the term related to viscosity due to electrolytic solutions having a viscosity close to the one of the water. In Eq. (31), the model obtained on the aforementioned conditions is shown.

$$Nu = 0.42 Re\_{a}^{0.694} Pr^{0.33} \tag{31}$$

Having mathematical software's backing, the exponents and the constant of Eq. (16) have been expressed in terms of more complex functions, decreasing the error generated by the model. For instance, Karcz and Streck [18] carried out experiments for continuous operations on water heating using steam, varying the type of impeller and its height in relation to the tanks bottom. With the data obtained and arbitrating the exponents values on Prandtl's number as 0.33 and on viscosity relation as 0.14, like Mohan et al. [19], Karcz and Streck [18] developed a polynomial for the K constant determination in Eq. (16) for each mechanical impeller used. Eq. (32) presents the model determined on the experiments.

$$Nu = KRe\_{a}{}^{0.67}Pr^{0.33} \left(\frac{\mu}{\mu\_w}\right)^{0.14} \tag{32}$$

In Eq. (33), the polynomial obtained from Eq. (32) with axial mechanical impeller to determine ho is illustrated.

$$K = 0.3119 + 0.3333 \times 10^{-3} \cdot \left(\frac{F - 6}{2}\right) + 1.75 \times 10^{-3} \left(\frac{G - 0.06}{0.015}\right) - 6.8333 \times 10^{-3} \left(\frac{Y - 0.184}{0.0694}\right)$$

$$+ 2.3333 \times 10^{-3} \left(\frac{X' - 0.0128}{0.0638}\right)$$

$$= 0.3333 \times 10^{-3} \tag{33}$$

where F, G, Y, and X are dimensional parameters, expressed in International System of Units relating to the tank and mechanical impeller.

Nassar and Mehrotra [20] present the coefficients in Eq. (16) for an isothermal heating equipped with a six-flat-blade turbine impeller, using condensed steam as heating fluid. This is shown in Eq. (34).

$$N\mu = 0.44 Re\_{a}^{0.67} Pr^{0.33} \left(\frac{\mu}{\mu\_w}\right)^{0.24} \tag{34}$$

#### 3.2. Helical coils

The helical coils present several advantages over jackets, because they have a large heat transfer area, are compacts, and, due to their geometry, promote an efficient turbulence on the fluids to be agitated in the tank. Moreover, it is an equipment that has low cost and it is easy to build [21]. The increase on the heat transfer through this kind of surface has two disadvantages: the need for baffles to avoid the formation of vortexes and the difficulty of maintenance and cleaning when submitted to fouling fluids. The helical coils can be single ribbon and double ribbon. The single ribbon coils are more efficient than the double ribbon ones, because as they have only one tube filament, there is a small formation of stationary zones, while for double ribbon coils, there is formation of stationary zones between the two tube filaments, decreasing the turbulence on this region and, as a consequence, the heat transference.

The characteristics of an efficient mixture with helical coils are consequence of developments of secondary flows throughout the pipeline, which is known as Dean's effect [22]. The secondary flows appear due to the action of centrifugal force on the fluid unit, being the difference of axial speed on the fluid unit for the tubes cross section as a function of a force gradient. The fluid elements in the tube center are projected perpendicularly to the tubes, that is, against the tank's wall, where they undergo a speed decrease, and when returning to the tube center, they find the main flows, resulting in vortexes throughout the helical coil. These vortexes are called Dean's cells, which are responsible for the increase of heat transfer efficiency by providing intensification on local turbulence close to the coil [23].

However, this effect is significant when in laminar and transient flows, because the vortexes' stability is kept, although, in turbulent flows, the vortexes end up being negligible in relation to the turbulence generated on the system.

Similarly with jackets, several researchers provided the exponents and the constant for Eq. (16) for single ribbon helical coils, which can also be used on double ribbon coils just adding safety factors.

Cummings and West [24] determined the exponents of Eq. (16) for organic liquids heating in a 0.76 m diameter tank equipped with a six-flat-blade radial impeller without baffles, on discontinuous operation. The obtained model is expressed in Eq. (35).

$$Nu = 1.01 Re\_a^{0.62} Pr^{0.33} \left(\frac{\mu}{\mu\_w}\right)^{0.14} \tag{35}$$

DeMaerteleire [25] carried out a similar study to the one done by Cummings and West [24]; however, he added four baffles to the tank and varied the turbine impeller diameter. A term was added to the obtained model, which relates the tank's diameter to the mechanical impeller diameter, thus obtaining a more realistic model that is valid for a Reynolds ranging from 26,000 to 110,000. The obtained expression is illustrated in Eq. (36).

Nassar and Mehrotra [20] present the coefficients in Eq. (16) for an isothermal heating equipped with a six-flat-blade turbine impeller, using condensed steam as heating fluid. This

The helical coils present several advantages over jackets, because they have a large heat transfer area, are compacts, and, due to their geometry, promote an efficient turbulence on the fluids to be agitated in the tank. Moreover, it is an equipment that has low cost and it is easy to build [21]. The increase on the heat transfer through this kind of surface has two disadvantages: the need for baffles to avoid the formation of vortexes and the difficulty of maintenance and cleaning when submitted to fouling fluids. The helical coils can be single ribbon and double ribbon. The single ribbon coils are more efficient than the double ribbon ones, because as they have only one tube filament, there is a small formation of stationary zones, while for double ribbon coils, there is formation of stationary zones between the two tube filaments,

decreasing the turbulence on this region and, as a consequence, the heat transference.

intensification on local turbulence close to the coil [23].

tinuous operation. The obtained model is expressed in Eq. (35).

Nu ¼ 1:01Rea

to the turbulence generated on the system.

adding safety factors.

The characteristics of an efficient mixture with helical coils are consequence of developments of secondary flows throughout the pipeline, which is known as Dean's effect [22]. The secondary flows appear due to the action of centrifugal force on the fluid unit, being the difference of axial speed on the fluid unit for the tubes cross section as a function of a force gradient. The fluid elements in the tube center are projected perpendicularly to the tubes, that is, against the tank's wall, where they undergo a speed decrease, and when returning to the tube center, they find the main flows, resulting in vortexes throughout the helical coil. These vortexes are called Dean's cells, which are responsible for the increase of heat transfer efficiency by providing

However, this effect is significant when in laminar and transient flows, because the vortexes' stability is kept, although, in turbulent flows, the vortexes end up being negligible in relation

Similarly with jackets, several researchers provided the exponents and the constant for Eq. (16) for single ribbon helical coils, which can also be used on double ribbon coils just

Cummings and West [24] determined the exponents of Eq. (16) for organic liquids heating in a 0.76 m diameter tank equipped with a six-flat-blade radial impeller without baffles, on discon-

0:62Pr0:33 μ

μw 0:14

0:67Pr0:33 μ

μw 0:24

(34)

(35)

Nu ¼ 0:44Rea

is shown in Eq. (34).

46 Heat Exchangers– Design, Experiment and Simulation

3.2. Helical coils

$$Nu = 1.778 Re\_a^{0.628} Pr^{0.33} \left(\frac{\mu}{\mu\_w}\right)^{0.20} \left(\frac{D\_t}{D\_a}\right)^{0.382} \tag{36}$$

The dimensionless group (Dt/Da) relates the tanks' diameter to the impeller's diameter; however, as demonstrated by Karcz and Streck [18], the best mixing conditions occur when the relation(Dt/Da) = 3. For instance, if the tank's diameter is 1 m, according to the standard condition, the impeller's diameter should be 0.33 m; replacing these values with the dimensionless group and powering it to 0.382, the value 1.527 is found. This value will be fixed regardless of the chosen tank's diameter, being the only request to follow the aforementioned standard condition. The influence of an increase or decrease on the tank's diameter and consequently the mechanical impeller diameter is evident on the flow regime (Reynolds) and on the heat transference (Nusselt).

Havas et al. [26] determined the exponents and the constant in Eq. (16) for water heating in a tank with 0.4 m of diameter and radial impeller with different diameters. The same experiments were carried out in a tank with 0.8 m of diameter. The model obtained by Havas et al. shows a greater coverage compared with the ones presented in Eqs. (35) and (36), because it takes into account a wider range of tank's diameter. Eq. (37) shows the determined model based on the obtained results.

$$Nu = 0.187 Re\_a^{0.688} Pr^{0.36} \left(\frac{\mu}{\mu\_w}\right)^{0.11} \left(\frac{D\_t}{D\_a}\right)^{0.62} \tag{37}$$

Yet, regarding Eq. (16), Couper et al. [27] recommended the Prandtl's number powered to a 0.37 exponent and the c exponent, which relates the fluid viscosities, determined by an empirical expression as function of the viscosity in the average inlet and outlet temperature of the fluid to be heated, as pointed by Eq. (38).

$$
\sigma = \frac{0.714}{\overline{\mu}^{0.21}} \tag{38}
$$

Dias et al. [28] determined the exponents and the constant in Eq. (16) for water heating equipped with an axial impeller with four flat blades inclined by 45° and a turbine radial impeller with six flat blades, based on Couper's assumption. The obtained models for axial and radial impellers are expressed by Eqs. (39) and (40), respectively.

$$Nu = 0.317. Re\_{\mu}^{0.589} Pr^{0.37} \left(\frac{\mu}{\mu\_w}\right)^{0.79} \tag{39}$$

$$Nu = 0.195 Re\_a^{0.664} Pr^{0.37} \left(\frac{\mu}{\mu\_w}\right)^{0.79} \tag{40}$$

#### 3.3. Spiral coils

Spiral coils are used, primarily, in tanks for storing heavy oil, where flow is not recommended when at ambient temperatures, due to the oil's high viscosity. As a solution, these coils are placed on the tank's base, where heating steam transfers energy to the oil, which increases oil's temperatures, making the viscosity on that spot decrease, allowing pumping transportation.

As these heat transfer surfaces meet in the vessel's bottom, even with intense turbulence generated by the mechanical impeller, it is evident that the heat transfer will occur predominantly on the tank's bottom and will spread on the axial axis over time. In view of this problem, Ho and Wijeysundera [29] simulated the heat transfer on the air dehumidification process with spiral coil and numerical simulations; they determined the internal heat transfer coefficient (hi) and, as a consequence, the heat transference performance throughout the process.

Naphon [30] determined a correlation for the hi coefficient in a vessel without mechanical impeller for cooling of hot water by using cold water. The model was based on Prandtl's number and Dean's number; however, the expression presented is of a totally particular nature to the experiment conditions used, not being possible, and then to extrapolate it for industrial units. The unit is operated in discontinuous state and natural convection, given that the obtained model is expressed in Eq. (41).

$$Nu = 27.358De^{0.287}Pr^{-0.949} \tag{41}$$

Rosa et al. [31] conducted a study as to predict the Nusselt's number when heating organic solutions in tanks equipped with spiral coils for two kinds of mechanical impellers, Rushton turbine (RT) with six flat blades and pitched blade turbine (PBT) with four blades at 45°. The obtained models are based in Eq. (16) and can be applied for Reynolds between 2000 and 500,000 and Prandtl around 3.8 and 140 and tanks containing baffles. The models for RT and PBT impellers are presented in Eqs. (42) and (43).

$$Nu = 0.10 Re\_a^{0.83} Pr^{0.33} \left(\frac{\mu}{\mu\_w}\right)^{0.14} \tag{42}$$

$$Nu = 0.81 Re\_{\mu}^{0.64} Pr^{0.33} \left(\frac{\mu}{\mu\_w}\right)^{0.14} \tag{43}$$

#### 3.4. Vertical tube baffles

The vertical tubular baffles consist of tube bank connected between themselves, which are located in the interior side of the tank, in the same way of vertical baffles used in tanks with jackets and coils. The dimensions to construct tube banks follow the standard agitation configurations determined by Rushton et al. [32], which states that the tube bank's diameter should have 1/10 of the tank's diameter in order to reach the best thermal and mixing efficiency.

Nu ¼ 0:317:Rea

Nu ¼ 0:195Rea

and, as a consequence, the heat transference performance throughout the process.

3.3. Spiral coils

48 Heat Exchangers– Design, Experiment and Simulation

obtained model is expressed in Eq. (41).

PBT impellers are presented in Eqs. (42) and (43).

3.4. Vertical tube baffles

0:589Pr0:37 μ

0:664Pr0:37 μ

Spiral coils are used, primarily, in tanks for storing heavy oil, where flow is not recommended when at ambient temperatures, due to the oil's high viscosity. As a solution, these coils are placed on the tank's base, where heating steam transfers energy to the oil, which increases oil's temperatures, making the viscosity on that spot decrease, allowing pumping transportation. As these heat transfer surfaces meet in the vessel's bottom, even with intense turbulence generated by the mechanical impeller, it is evident that the heat transfer will occur predominantly on the tank's bottom and will spread on the axial axis over time. In view of this problem, Ho and Wijeysundera [29] simulated the heat transfer on the air dehumidification process with spiral coil and numerical simulations; they determined the internal heat transfer coefficient (hi)

Naphon [30] determined a correlation for the hi coefficient in a vessel without mechanical impeller for cooling of hot water by using cold water. The model was based on Prandtl's number and Dean's number; however, the expression presented is of a totally particular nature to the experiment conditions used, not being possible, and then to extrapolate it for industrial units. The unit is operated in discontinuous state and natural convection, given that the

Rosa et al. [31] conducted a study as to predict the Nusselt's number when heating organic solutions in tanks equipped with spiral coils for two kinds of mechanical impellers, Rushton turbine (RT) with six flat blades and pitched blade turbine (PBT) with four blades at 45°. The obtained models are based in Eq. (16) and can be applied for Reynolds between 2000 and 500,000 and Prandtl around 3.8 and 140 and tanks containing baffles. The models for RT and

0:83Pr0:33 μ

0:64Pr0:33 μ

The vertical tubular baffles consist of tube bank connected between themselves, which are located in the interior side of the tank, in the same way of vertical baffles used in tanks with jackets and coils. The dimensions to construct tube banks follow the standard agitation

μw 0:14

μw 0:14

Nu ¼ 0:10Rea

Nu ¼ 0:81Rea

μw 0:79

μw 0:79

Nu ¼ 27:358De0:287Pr�0:949 (41)

(39)

(40)

(42)

(43)

According to Tatterson [33], the vertical tubular baffles increases 37% the heat transfer efficiency on the turbulent region; this happens due to the large heat transfer area of tubular baffles and the intense turbulence caused by the mechanical impeller, in such a way that tanks with this kind of surface do not need additional baffles for vortex breaking, since the very thermal surface also works as a baffle.

Studies carried out by Rosa et al. [34] with water and sucrose solutions heating in tanks equipped with axial and radial impellers determined that the rotation in which the heat transfer is maximized with the lowest energy usage by mechanical impeller ranges from 200 to 300 rpm.

The external convection coefficient of heat transfer (ho) is a function of several variables, which have already been mentioned on item 1; however, when using vertical tubular baffles, this coefficient is also reliant on the amount of bench tubes used, on their spatial placement in the tank's interior, and on the kind of operation (continuous or discontinuous) [35].

The exponents and the constant in Eq. (16) were not generically determined for any tank dimension, vertical tubular baffles, and amount of tube banks, due to the complexity presented in the previous paragraph, that is, most part of the models presented on today's literature are for discontinuous operations.

The first model determined in order to estimate the ho coefficient using vertical tubular baffles was presented on the pioneer work carried out by Rushton et al. [17] in tanks abiding by standard dimensions. The experiments were ran in a 0.366 m diameter tank where hot water was used to heat oil through four vertical tube bank using a turbine impeller with six flat blades and a radial impeller with four flat blades. The operation was performed as a discontinuous operation.

The models obtained by Rushton et al. were found by plotting directly the experimental values of the ho coefficient in function of the variables within the Reynolds number modified for agitated tanks, which are the impeller's diameter, the mechanical impeller's rotation, the density and the viscosity of the fluid in study.

The models for turbine radial impellers with six flat blades and with four flat blades are represented in Eqs. (44) and (45), respectively.

$$h\_{\vartheta} = 0.00285 D^2 N \rho \mu^{-1} \tag{44}$$

$$h\_o = 0.00235 D^{1.1} N^{0.7} \rho^{0.7} \mu^{-0.7} \tag{45}$$

Dunlap and Rushton [36] continued the study carried by Rushton et al. [17] with vertical tube baffles, incorporating to the obtained model the effect of the impeller's diameter in relation to the tank's diameter and the effect of the amount of tube bank (nb) on the heat transfer from hot water to oils. Eq. (46) presents the determined model.

$$Nu = 0.09 Re\_a^{0.65} Pr^{0.33} \left(\frac{\mu}{\mu\_w}\right)^{0.4} \left(\frac{D\_a}{D\_t}\right)^{0.33} \left(\frac{2}{n\_b}\right)^{0.2} \tag{46}$$

Havas et al. [37] carried out experiments in a tank with 0.4 and 0.8 m of diameter for water and fuel oils heating with five-tube bank and radial impeller equipped with six flat blades. The operation was conducted on discontinuous mode. Based on the experimental data, the researchers concluded that the effect caused by the impellers' diameter and by the amount of tube bank is negligible in relation to the turbulence generated by the mechanical impeller. The obtained model incorporates the effects, which are directly disregarded on the Reynolds number, as presented in Eq. (47).

$$Nu = 0.208 Re\_{\mu}^{0.65} Pr^{0.33} \left(\frac{\mu}{\mu\_w}\right)^{0.4} \tag{47}$$

Karcz and Strek [38] defined a model to determine the external convection coefficients using an axial mechanical impeller with three inclined blades in nonstandard conditions for the vertical tubular baffle dimensions. The model is presented in Eq. (48).

$$Nu = 0.494 Re\_d^{0.67} Pr^{0.33} \left(\frac{\mu}{\mu\_w}\right)^{0.14} \tag{48}$$

Lukes [39] expanded the studies carried by Karcz and Strek [38], using an axial impeller with three inclined blades and a standard axial impeller with four blades inclined 45°, obtaining an expression similar to Eq. (48). The obtained function is presented in Eq. (49).

$$Nu = 0.542 Re\_d^{0.65} Pr^{0.33} \left(\frac{\mu}{\mu\_w}\right)^{0.40} \tag{49}$$

Rosa et al. [40] carried experiments in a tank with 0.4 m diameter for heating sucrose solutions at 20% and 32% (concentration?) using hot water, through four-tube bank using an axial impeller with four blades inclined 45°; however, this is the first model presented operating in continuous operation, as illustrated in Eq. (50).

$$Nu = 17.88 Re\_d^{0.27} Pr^{0.29} \left(\frac{\mu}{\mu\_w}\right)^{0.37} \tag{50}$$

Rosa et al. [41] carried out experiments under the same conditions that Rosa et al. [35], however, this time using a turbine radial impeller. The model is shown in Eq. (51).

$$Nu = 25.03 Re\_a^{0.38} Pr^{0.11} \left(\frac{\mu}{\mu\_w}\right)^{0.20} \tag{51}$$


**2.2 Solar cell fabrication techniques**

2 Will-be-set-by-IN-TECH

nanomorphology of polymeric solar cells plays a crucial role for the performance of the

Historically, thermal annealing of the film has been used to induce the phase separation between donor and acceptor in bulk-heterojunction blends.(Padinger et al., 2003) However, thermal treatment creates an additional fabrication step in the whole device fabrication process. Later, various methods have been tested and employed to control the nanomorphology of the blends, namely use of solvents with different boiling points (choice of solvent), reduction of drying speed (rate of drying and vapor annealing), changing the solubility of materials, melting of bilayers and the use of processing additives.(Pivrikas et al., 2010b) The later method has received great academic interest as it removes the need for post-production treatment while at the same time allowing fine control of the

In this work the factors limiting the power conversion efficiency of excitonic polymer-based bulk-heterojunction solar cells are discussed. Various methods allowing the film nanomorphology to be controlled are reviewed. The use of processing additives to control the phase separation for the formation of an interpenetrating network and how this impacts

Polymer-based bulk-heterojunction solar cells (BHSC) have already shown certified efficiencies above 8 % demonstrating ability to compete with inorganic solar cell systems (eg. amorphous silicon cells fabricated on flexible substrates). Efficiencies exceeding 10% for solution processed solar cells are expected to be achieved soon.(Nayak et al., 2011) The power conversion efficiency of BHSCs is determined by the photophysical processes under operational conditions. A fundamental understanding of the relation between light absorption charge separation, charge transport, recombination, and film nanostructure as well as between the various thin film fabrication and processing parameters (such as solvent composition, solution concentration, deposition atmosphere and process temperature) is needed for further improvements. These important parameters can be controlled to some extent by adjusting the required film composition or device structure.(Gunes et al., 2007)

The typical device structure of most organic optoelectronic devices, including organic light-emitting diodes and solar cells, is shown in Fig. 1. The front electrode is based on a transparent conducting oxide, such as indium tin oxide (ITO), that serves as the high-work-function, positive electrode.(Brabec et al., 2001) To further improve the quality of the ITO electrode and aid hole (positive charge carrier) extraction from the active film, poly(3,4-ethylenedioxythiophene)-poly(styrene sulfonate) (PEDOT-PSS) layer (tens of nanometers thick) is coated on top, forming a smooth surface which is essential in thin film devices. The photoactive film, the donor and acceptor bulk-heterojunction blend is deposited on top of the PEDOT-PSS layer. The whole device is finished by thermally evaporating the back contact (negative electrode) under high vacuum. To achieve the built-in electric field needed for most devices to operate, the back electrode must be made from a low-work-function metal that serves as the negative electrode. In the operation of a typical polymer-fullerene bulk-heterojunction solar cell, electrons generated in the active layer are collected by the back electrode (anode), and holes are collected at the opposite electrode

nanomorphology in various donor-acceptor blends.(Lee et al., 2008)

the power conversion efficiency is described.

**2. Excitonic polymer-based solar cells**

**2.1 Solar cell device structure**

(cathode).

devices.

While academic research is highly concentrated on improving the power conversion efficiency, there are other important aspects needed for commercial success, such as cell stability, degradation, low manufacturing costs with rapid large scale production. This has been summarized by the Venn diagram as the unification of challenges when trying to combine power conversion efficiency, processability and stability into final devices.(Jorgensen et al., 2008)

Solution processing is attractive for fabricating organic optoelectronic devices mainly due to its simplicity and applicability for large scale and low-cost production. Thin films can be formed in various ways: a) printing techniques including screen printing, pad printing, gravure printing, flexographic printing and offset printing; b) coating techniques including pin coating, doctor blading, casting, painting, spray coating, slot-die coating, curtain coating, slide coating and knife-over-edge coating. The only technique that in both categories is inkjet printing.

Spin coating has been the most common technique for polymeric solar cell fabrication with numerous reviews and fundamental studies available.(Norrman et al., 2005) This technique, widely used in the microelectronics industry to deposit photoresist on silicon wafers, allows for the reproducible formation of highly homogeneous films over large areas. A typical spin coating process involves application of a solution (with the organic semiconductors dissolved in a solvent) to a substrate which is then either accelerated to the required angular velocity or is already spinning at it, Fig. 2.(Krebs, 2009) A large portion of the solution is wasted leaving a thin film on the substrate. Film thickness, morphology and surface topography strongly depend on the rotational speed, viscosity, volatility, diffusivity, molecular weight and concentration of the solutes and solvents used.(Cohen & Gutoff, 1992)

### **2.3 Current-Voltage dependence of solar cells**

The most important figure of merit describing the performance of a solar cell is the power conversion efficiency, which is determined from the current voltage characteristics of the solar cells under operational conditions. Typical current-voltage characteristics of solar cells under illumination is shown in Fig. 3.(Deibel & Dyakonov, 2010)

The accurate measurement of the PCE according to international standards has been described in the literature,(Shrotriya et al., 2006) and is eesential for reproducibility and comparison of results between different laboratories. The Shockley diode equation describes the

Fig. 1. Schematic sandwich-type structure of organic solar cells showing an organic semiconductor active film between two metal electrodes with different work functions (typically ITO/PEDOT-PSS as positive and Ca/Al as negative contacts). Reprinted with permission from (Shaheen, 2007). Copyright 2007 Society of Photo-Optical Instrumentation Engineers.

Fig. 3. Typical schematic current-voltage (I-V) dependence of the organic solar cell in the dark and under illumination. Reprinted with permission from (Deibel & Dyakonov, 2010).

Relation Between Nanomorphology and Performance of Polymer-Based Solar Cells 125

PCE is defined as the ratio of the electrical power produced by a solar cell to the optical power

*PCE* = *jSCVOCFF Plight*

where *jSC* is the short circuit current density, *VOC* is the open circuit voltage, *FF* = *VMPjmp*/*VOCjSC* is the fill factor representing the maximum area in the fourth quadrant of the I-V characteristics of solar cells, *VMP* and jmp are the voltage and current, respectively, at the point of maximum power, and *Plight* is the power of incoming light under Standard Test Conditions (1000 W/m2, AM 1.5 (Air Mass) solar reference spectrum, temperature during measurements 25 C).[15] To maximize the PCE values, all these parameters have to be maximized. The open circuit voltage of BHSC is determined by the energy of the quasi-Fermi levels of both semiconductors (donor and acceptor) as well as the Fermi levels of the electrodes.(Scharber et al., 2006) The short circuit current *jSC* depends on electrical carrier drift (induced by electric field) and diffusion (induced by concentration gradient). Apart from the absorption on the film, charge carrier concentration and mobility are the main factors influencing the photocurrent regardless of the transport mechanism (drift or

(1)

Copyright 2010, with permission from Institute of Physics.

of incident light (*Plight*):(Luque & Hegedus, 2003)

**2.4 Power conversion efficiency**

diffusion).(Nelson, 2003)

Fig. 2. Spin coating of organic solar cells from solution. Reprinted with permission from (Krebs, 2009). Copyright 2009, with permission from Elsevier.

current-voltage dependence of an ideal diode.(Shockley & Queisser, 1961) In the dark under forward bias the injection current increases exponentially with applied bias, whereas under reverse bias, current saturates at low applied voltages due to blocking contacts. This leads to a rectifying behaviour as can be clearly seen in the log-lin plot in Fig. 3. A description of the non-ideal device (typical organic solar cells) requires addition of series (Rs) and paralell (Rp) resistances. Rs is connected in series with the ideal diode and it describes the contact resistances such as injection barriers and sheet resistances. Rp arises due to the influence of local shunts between the two electrodes, i.e. additional current paths circumventing the diode. Typically in organic solar cells, a strong photocurrent dependence of applied electric field is observed manifesting as a non-saturated current at -1V reverse bias (Fig. 3 ). The field-dependent photocurrent arises due to:

a) field dependent mobile charge carrier generation, since an exciton has to dissociate into mobile carriers;(Oesterbacka et al., 2010)

b) charge carrier collection due to Hecht's law, if the extracted charge saturates at electric fields wheere the film thickness is larger than the carrier drift distance.(Hecht, 1932)

c) electric field dependent carrier mobility.(Pivrikas, Ullah, Sitter & Sariciftci, 2011)

In addition to electric field dependent mobility, the charge transport in disordered organic solar cells is also carrier concentration dependent. This effect arises due to the hopping nature of charge transport, where at higher carrier concentrations the carrier hopping probability between localized states increases (loosely speaking due to a higher density of localized states resulting in better electron wavefunction overlap) and therefore the carrier mobility increases.(PIVRIKAS, ULLAH, SINGH, SIMBRUNNER, MATT, SITTER & SARICIFTCI, 2011)

Fig. 3. Typical schematic current-voltage (I-V) dependence of the organic solar cell in the dark and under illumination. Reprinted with permission from (Deibel & Dyakonov, 2010). Copyright 2010, with permission from Institute of Physics.

#### **2.4 Power conversion efficiency**

4 Will-be-set-by-IN-TECH

Fig. 2. Spin coating of organic solar cells from solution. Reprinted with permission from

current-voltage dependence of an ideal diode.(Shockley & Queisser, 1961) In the dark under forward bias the injection current increases exponentially with applied bias, whereas under reverse bias, current saturates at low applied voltages due to blocking contacts. This leads to a rectifying behaviour as can be clearly seen in the log-lin plot in Fig. 3. A description of the non-ideal device (typical organic solar cells) requires addition of series (Rs) and paralell (Rp) resistances. Rs is connected in series with the ideal diode and it describes the contact resistances such as injection barriers and sheet resistances. Rp arises due to the influence of local shunts between the two electrodes, i.e. additional current paths circumventing the diode. Typically in organic solar cells, a strong photocurrent dependence of applied electric field is observed manifesting as a non-saturated current at -1V reverse bias (Fig. 3 ). The

a) field dependent mobile charge carrier generation, since an exciton has to dissociate into

b) charge carrier collection due to Hecht's law, if the extracted charge saturates at electric fields

In addition to electric field dependent mobility, the charge transport in disordered organic solar cells is also carrier concentration dependent. This effect arises due to the hopping nature of charge transport, where at higher carrier concentrations the carrier hopping probability between localized states increases (loosely speaking due to a higher density of localized states resulting in better electron wavefunction overlap) and therefore the carrier mobility increases.(PIVRIKAS, ULLAH, SINGH, SIMBRUNNER, MATT, SITTER & SARICIFTCI, 2011)

wheere the film thickness is larger than the carrier drift distance.(Hecht, 1932) c) electric field dependent carrier mobility.(Pivrikas, Ullah, Sitter & Sariciftci, 2011)

(Krebs, 2009). Copyright 2009, with permission from Elsevier.

field-dependent photocurrent arises due to:

mobile carriers;(Oesterbacka et al., 2010)

PCE is defined as the ratio of the electrical power produced by a solar cell to the optical power of incident light (*Plight*):(Luque & Hegedus, 2003)

$$PCE = \frac{j\_{\text{SC}}V\_{\text{OC}}FF}{P\_{\text{light}}} \tag{1}$$

where *jSC* is the short circuit current density, *VOC* is the open circuit voltage, *FF* = *VMPjmp*/*VOCjSC* is the fill factor representing the maximum area in the fourth quadrant of the I-V characteristics of solar cells, *VMP* and jmp are the voltage and current, respectively, at the point of maximum power, and *Plight* is the power of incoming light under Standard Test Conditions (1000 W/m2, AM 1.5 (Air Mass) solar reference spectrum, temperature during measurements 25 C).[15] To maximize the PCE values, all these parameters have to be maximized. The open circuit voltage of BHSC is determined by the energy of the quasi-Fermi levels of both semiconductors (donor and acceptor) as well as the Fermi levels of the electrodes.(Scharber et al., 2006) The short circuit current *jSC* depends on electrical carrier drift (induced by electric field) and diffusion (induced by concentration gradient). Apart from the absorption on the film, charge carrier concentration and mobility are the main factors influencing the photocurrent regardless of the transport mechanism (drift or diffusion).(Nelson, 2003)

materials (relative static permittivity is around 3) the primary photoexcitation is an exciton, which does not create the photocurrent.(Luque & Hegedus, 2003) The exciton diffusion length describes how far an exciton can diffuse within its lifetime. The concept of heterojunction between two organic semiconductors (donor and acceptor) is used to split an exciton into mobile charge carriers. Efficient exciton dissociation (charge transfer) takes place at the interface between donor and acceptor if a suitable offset in energy level exists, as shown in Fig. 4 b).(Pivrikas et al., 2010a) The excited state Charge Transfer (CT) complex (sometimes called exciplex) might be formed after the dissociation of an exciton meaning that positive and negative charge carriers might remain bounded by the Coulomb attraction at the donor

Relation Between Nanomorphology and Performance of Polymer-Based Solar Cells 127

The charge carrier transport (collection) to the electrodes is the third important limiting processes. As shown in Fig. 4 c), mobile electrons and holes must be transported to the opposite electrodes. The driving force can be either diffusion, related to the carrier concentration gradients, and/or drift due to a built-in electric field. An important aspects of charge transport are the charge extraction at the semiconductor-metal interface. The energy level alignment between the metal and semiconductor, free charge carrier concentration in the film (doping level) as well as the trapping level concetration, carrier capture and release times from capture centers determine the interfacial properties of the device. Non-blocking contact without energetical barrier for charge carrier extraction is required to be present at the interface.(Baranovski, 2006) The disordered nature of solution processed films of organic semiconductors results in low charge carrier mobilities (tzpically 10−3 - 10−7 cm2V−1s−1 in *π*-conjugated polymers). The mobility of the slower charge carrier limits the photocurrent, and therefore the efficiency of the solar cell due to accumulation of charge carriers. Since the photocurrent under operation conditions typically approach space charge limited current, second order recombination processes become dominant due to high charge carrier concentration, and the carrier lifetime becomes shorter than the transit time.(Pivrikas

The light absorption coefficient, *α*, in a disordered organic film is usually high, on the order of 105 cm−1. This allows thin films, on the order of hundreds of nanometers, to be used in solar cells. However, the exciton diffusion length in most organic materials is of the order of 10 nm. If the exciton is to diffuse to the interface between the two materials (donor-acceptor) in order to separate into mobile charge carriers, these two materials must be blended on this length scale. Furthermore, the donor-acceptor phases must for bi-continuous network with percolating pathways for electron and hole transport to the elctrodes. This is the operating principle of the BHSC shown in Fig. 5.(Sariciftci, 2006) The film nano-morphology is crucially important for the efficiency of solar cells.(Ma et al., 2005) The nanoscale phase-separation phase separation between donor and acceptor in BHSC plays an important role relating the device properties and performance to the solar cell fabrication methods. Typical donor is poly(3-hexylthiophene-2,5-diyl) (P3HT) and acceptor is [6,6]-phenyl-C61-butyric acid methyl

The formation and the size of nanoscale domains of donor and acceptor phases are strongly dependent on the film fabrication techniques and conditions. Beyond the selection of suitable materials there are several parameters that must be carefully controlled when fabricating

acceptor interface, which would not contribute to photocurrent.

et al., 2010a)

ester (PCBM)

**2.6 Bulk-heterojunction solar cells**

**3. Methods to control the morphology of BHSC**

Fig. 4. Factors limiting power conversion efficiency of excitonic solar cells. Reprinted with permission from (Pivrikas, 2010). Copyright 2010 IEEE.

The PCE of organic solar cells is influenced by many different photophysical processes and parameters. Simplified energy-level diagrams of organic solar cells utilizing donor and acceptor materials is shown in Fig. 2.(Pivrikas et al., 2010a)

### **2.5 Power conversion efficiency limiting mechanisms in excitonic solar cells**

The first factor limiting the PCE is the absorption of light in the film. Ideally, as much as possible of the incident solar irradiance should be absorbed.(Pivrikas et al., 2010a) The Beer-Lambert law determines the light absorption profile in homogeneously distributed and scatter-free medium. Optical interference effects can also influence the light absorption profile in thin multilayer films.(Dennler et al., 2009) As can be seen in Fig. 4 some part of absorbed light energy is lost due to a thermalization process - charge relaxtion within the Density of States (DOS) to the lower energy levels to form an occupational-DOS within localized DOS.(Bassler, 1993; Juska et al., 2003; Osterbacka et al., 2003)

The second efficiency limiting process is exciton dissociation into mobile charge carriers. Due to low dielectric constants and consequently weak Coulombic field screening in the organic materials (relative static permittivity is around 3) the primary photoexcitation is an exciton, which does not create the photocurrent.(Luque & Hegedus, 2003) The exciton diffusion length describes how far an exciton can diffuse within its lifetime. The concept of heterojunction between two organic semiconductors (donor and acceptor) is used to split an exciton into mobile charge carriers. Efficient exciton dissociation (charge transfer) takes place at the interface between donor and acceptor if a suitable offset in energy level exists, as shown in Fig. 4 b).(Pivrikas et al., 2010a) The excited state Charge Transfer (CT) complex (sometimes called exciplex) might be formed after the dissociation of an exciton meaning that positive and negative charge carriers might remain bounded by the Coulomb attraction at the donor acceptor interface, which would not contribute to photocurrent.

The charge carrier transport (collection) to the electrodes is the third important limiting processes. As shown in Fig. 4 c), mobile electrons and holes must be transported to the opposite electrodes. The driving force can be either diffusion, related to the carrier concentration gradients, and/or drift due to a built-in electric field. An important aspects of charge transport are the charge extraction at the semiconductor-metal interface. The energy level alignment between the metal and semiconductor, free charge carrier concentration in the film (doping level) as well as the trapping level concetration, carrier capture and release times from capture centers determine the interfacial properties of the device. Non-blocking contact without energetical barrier for charge carrier extraction is required to be present at the interface.(Baranovski, 2006) The disordered nature of solution processed films of organic semiconductors results in low charge carrier mobilities (tzpically 10−3 - 10−7 cm2V−1s−1 in *π*-conjugated polymers). The mobility of the slower charge carrier limits the photocurrent, and therefore the efficiency of the solar cell due to accumulation of charge carriers. Since the photocurrent under operation conditions typically approach space charge limited current, second order recombination processes become dominant due to high charge carrier concentration, and the carrier lifetime becomes shorter than the transit time.(Pivrikas et al., 2010a)

### **2.6 Bulk-heterojunction solar cells**

6 Will-be-set-by-IN-TECH

Fig. 4. Factors limiting power conversion efficiency of excitonic solar cells. Reprinted with

The PCE of organic solar cells is influenced by many different photophysical processes and parameters. Simplified energy-level diagrams of organic solar cells utilizing donor and

The first factor limiting the PCE is the absorption of light in the film. Ideally, as much as possible of the incident solar irradiance should be absorbed.(Pivrikas et al., 2010a) The Beer-Lambert law determines the light absorption profile in homogeneously distributed and scatter-free medium. Optical interference effects can also influence the light absorption profile in thin multilayer films.(Dennler et al., 2009) As can be seen in Fig. 4 some part of absorbed light energy is lost due to a thermalization process - charge relaxtion within the Density of States (DOS) to the lower energy levels to form an occupational-DOS within localized

The second efficiency limiting process is exciton dissociation into mobile charge carriers. Due to low dielectric constants and consequently weak Coulombic field screening in the organic

**2.5 Power conversion efficiency limiting mechanisms in excitonic solar cells**

permission from (Pivrikas, 2010). Copyright 2010 IEEE.

acceptor materials is shown in Fig. 2.(Pivrikas et al., 2010a)

DOS.(Bassler, 1993; Juska et al., 2003; Osterbacka et al., 2003)

The light absorption coefficient, *α*, in a disordered organic film is usually high, on the order of 105 cm−1. This allows thin films, on the order of hundreds of nanometers, to be used in solar cells. However, the exciton diffusion length in most organic materials is of the order of 10 nm. If the exciton is to diffuse to the interface between the two materials (donor-acceptor) in order to separate into mobile charge carriers, these two materials must be blended on this length scale. Furthermore, the donor-acceptor phases must for bi-continuous network with percolating pathways for electron and hole transport to the elctrodes. This is the operating principle of the BHSC shown in Fig. 5.(Sariciftci, 2006) The film nano-morphology is crucially important for the efficiency of solar cells.(Ma et al., 2005) The nanoscale phase-separation phase separation between donor and acceptor in BHSC plays an important role relating the device properties and performance to the solar cell fabrication methods. Typical donor is poly(3-hexylthiophene-2,5-diyl) (P3HT) and acceptor is [6,6]-phenyl-C61-butyric acid methyl ester (PCBM)

### **3. Methods to control the morphology of BHSC**

The formation and the size of nanoscale domains of donor and acceptor phases are strongly dependent on the film fabrication techniques and conditions. Beyond the selection of suitable materials there are several parameters that must be carefully controlled when fabricating

enhanced as a result of phase separation between the donor and acceptor on meso (> 100 nm) and nanoscales (< 20 nm).(Kim, Cook, Tuladhar, Choulis, Nelson, Durrant, Bradley, Giles,

Relation Between Nanomorphology and Performance of Polymer-Based Solar Cells 129

The effect of thermal annealing on film morhology was clearly demonstrated by bright-field (BF) transmission electron microscopy (TEM) images, recorded in slight defocussing

Fig. 6. Transmission Electron Microscopy (TEM) images show the overview (a) and zoomed in (b), and the corresponding schematic representation (c) of the photoactive layer solar cells.

As can be seen, fibrillar-like P3HT domains (brighter in contrast compared to background) overlap with each other over the whole film. The inset of Fig. 6 shows the selected area electron diffraction (SAED) pattern of the film. The outer ring corresponds to a distance of 0.39 nm, which is typical pi-pi stacking distance of P3HT chains.(Ihn et al., 1993) The crystallinity of P3HT crystals is not very pronounced as seen from the low intensity of the reflection ring. The inner ring in the SAED pattern, corresponding to a d-spacing of 0.46 nm, is seen to be even more diffuse and has been attributed by the nanometer sized PCBM nanocrystals that are homogeneously dispersed throughout the film. Fig. 6 a) shows the BF TEM images of the composite film after annealing (120 C for 60 min). The most pronounced feature in the BF TEM image of the annealed sample is the increased contrast and the appearance of bright fibrillar P3HT crystals throughout the entire film. The width of these crystals remains almost constant compared to the pristine composite film, but on average their length was found to increase over 50 %. The increased crystallinity of P3HT after thermal treatment

Left (a)-(c) images: pristine unannealed P3HT and PCBM blend. Right (a)-(c) images: thermal annealed P3HT and PCBM blend. Insets in (a) figures is the corresponding SAED pattern. The dash line bordered regions represent the extension of existing P3HT crystals in the pristine film or newly developed PCBM-rich domains during the annealing step. The arrow is to indicate the increased intensity of (020) Debye-Scherrer ring from P3HT crystals compared to the SAED pattern shown in the inset of Figure 2a. Reprinted with permission

from (Yang et al., 2005). Copyright 2005 American Chemical Society.

conditions in a P3HT/PCBM blends, as shown in Fig. 6.(Yang et al., 2005)

McCulloch, Ha et al., 2006; Ma et al., 2005)

Fig. 5. Bilayer and bulk-heterojunction solar cells. Reprinted with permission from (Pivrikas et al., 2010a). Copyright 2009, with permission from Elsevier.

BHSC, such as the solution concentration, deposition temperature, donor-acceptor blend ratio, spin speeds using solvents with different boiling points, solvent evaporation kinetics, vapour pressure, solubility, and polarity.(Pivrikas et al., 2010b)

Various methods allowing the control of the film nano-morphology have been introduced in the past.(Chen, Hong, Li & Yang, 2009; Peet et al., 2009) Initially it was observed that the PCE of BHSC significantly increases upon a postproduction treatment, e.g. thermal annealing of solar cells with applied external voltage.(Hoppea & Sariciftci, 2004; Padinger et al., 2003) Other methods used in the past to control the morphology of BHSC involve the use of appropriate solvents with specific boiling point that allow either the increase or decrease of the solvent evaporation rate.(Kim, Choulis, Nelson, Bradley, Cook & Durrant, 2005; Shaheen et al., 2001; Yu et al., 1995) Other methods, such as reducing the drying speed of spin-coated films,(Li et al., 2005; Mihailetchi et al., 2006; Vanlaeke et al., 2006) solubility matching(Troshin et al., 2009) and the melting of bilayers have also been used.(Kim, Liu & Carroll, 2006) It was observed that chemical additives can substitute the post production treatment of BHSC.(Lee et al., 2008) Processing additives are an attractive concept due to the simplicity and suitability for large scale production.

### **3.1 Thermal annealing of devices**

Thermal annealing, by controlling the temperature and annealing time, is typically applied to either the final device or BHJ films in order to improve the nanoscale phase separation between donor and acceptor.(Ma et al., 2005; Sun et al., 2007; Xin et al., 2008; Zhang, Choi, Haliburton, Cleveland, Li, Sun, Ledbetter & Bonner, 2006) Significant improvement in photovoltaic performance after annealing is typically observed in P3HT/PCBM blends.(Padinger et al., 2003) Thermal annealing has the advantage in that it can be applied independently of the film deposition technique. Thermal annealing has also been shown to enhance the crystallinity of the polymer, such as for P3HT, increasing the PCE and the photocurrent due to increased carrier mobility.(Erb et al., 2005) Furthermore, the interconnections between the polymer/fullerene phazes in the interpenetrating network are 8 Will-be-set-by-IN-TECH

Fig. 5. Bilayer and bulk-heterojunction solar cells. Reprinted with permission from (Pivrikas

BHSC, such as the solution concentration, deposition temperature, donor-acceptor blend ratio, spin speeds using solvents with different boiling points, solvent evaporation kinetics, vapour

Various methods allowing the control of the film nano-morphology have been introduced in the past.(Chen, Hong, Li & Yang, 2009; Peet et al., 2009) Initially it was observed that the PCE of BHSC significantly increases upon a postproduction treatment, e.g. thermal annealing of solar cells with applied external voltage.(Hoppea & Sariciftci, 2004; Padinger et al., 2003) Other methods used in the past to control the morphology of BHSC involve the use of appropriate solvents with specific boiling point that allow either the increase or decrease of the solvent evaporation rate.(Kim, Choulis, Nelson, Bradley, Cook & Durrant, 2005; Shaheen et al., 2001; Yu et al., 1995) Other methods, such as reducing the drying speed of spin-coated films,(Li et al., 2005; Mihailetchi et al., 2006; Vanlaeke et al., 2006) solubility matching(Troshin et al., 2009) and the melting of bilayers have also been used.(Kim, Liu & Carroll, 2006) It was observed that chemical additives can substitute the post production treatment of BHSC.(Lee et al., 2008) Processing additives are an attractive concept due to the simplicity and suitability for large

Thermal annealing, by controlling the temperature and annealing time, is typically applied to either the final device or BHJ films in order to improve the nanoscale phase separation between donor and acceptor.(Ma et al., 2005; Sun et al., 2007; Xin et al., 2008; Zhang, Choi, Haliburton, Cleveland, Li, Sun, Ledbetter & Bonner, 2006) Significant improvement in photovoltaic performance after annealing is typically observed in P3HT/PCBM blends.(Padinger et al., 2003) Thermal annealing has the advantage in that it can be applied independently of the film deposition technique. Thermal annealing has also been shown to enhance the crystallinity of the polymer, such as for P3HT, increasing the PCE and the photocurrent due to increased carrier mobility.(Erb et al., 2005) Furthermore, the interconnections between the polymer/fullerene phazes in the interpenetrating network are

et al., 2010a). Copyright 2009, with permission from Elsevier.

pressure, solubility, and polarity.(Pivrikas et al., 2010b)

scale production.

**3.1 Thermal annealing of devices**

enhanced as a result of phase separation between the donor and acceptor on meso (> 100 nm) and nanoscales (< 20 nm).(Kim, Cook, Tuladhar, Choulis, Nelson, Durrant, Bradley, Giles, McCulloch, Ha et al., 2006; Ma et al., 2005)

The effect of thermal annealing on film morhology was clearly demonstrated by bright-field (BF) transmission electron microscopy (TEM) images, recorded in slight defocussing conditions in a P3HT/PCBM blends, as shown in Fig. 6.(Yang et al., 2005)

Fig. 6. Transmission Electron Microscopy (TEM) images show the overview (a) and zoomed in (b), and the corresponding schematic representation (c) of the photoactive layer solar cells. Left (a)-(c) images: pristine unannealed P3HT and PCBM blend. Right (a)-(c) images: thermal annealed P3HT and PCBM blend. Insets in (a) figures is the corresponding SAED pattern. The dash line bordered regions represent the extension of existing P3HT crystals in the pristine film or newly developed PCBM-rich domains during the annealing step. The arrow is to indicate the increased intensity of (020) Debye-Scherrer ring from P3HT crystals compared to the SAED pattern shown in the inset of Figure 2a. Reprinted with permission from (Yang et al., 2005). Copyright 2005 American Chemical Society.

As can be seen, fibrillar-like P3HT domains (brighter in contrast compared to background) overlap with each other over the whole film. The inset of Fig. 6 shows the selected area electron diffraction (SAED) pattern of the film. The outer ring corresponds to a distance of 0.39 nm, which is typical pi-pi stacking distance of P3HT chains.(Ihn et al., 1993) The crystallinity of P3HT crystals is not very pronounced as seen from the low intensity of the reflection ring. The inner ring in the SAED pattern, corresponding to a d-spacing of 0.46 nm, is seen to be even more diffuse and has been attributed by the nanometer sized PCBM nanocrystals that are homogeneously dispersed throughout the film. Fig. 6 a) shows the BF TEM images of the composite film after annealing (120 C for 60 min). The most pronounced feature in the BF TEM image of the annealed sample is the increased contrast and the appearance of bright fibrillar P3HT crystals throughout the entire film. The width of these crystals remains almost constant compared to the pristine composite film, but on average their length was found to increase over 50 %. The increased crystallinity of P3HT after thermal treatment

is oriented parallel - which is the typically observed P3HT orientation. Upon annealing the as-prepared films at various temperatures, the d-spacing along the *a*-axis of the P3HT crystal was found to remain constant, indicating that during the interdiffusion process, the PCBM does not interpenetrate between the side chains of the P3HT crystal structure.(Mayer et al., 2009) The peak width of the diffraction ring, corresponding to the aggregates of PCBM does not change during the interdiffusion process, showing that PCBM remains in an amorphous state with aggregates large enough to scatter incident X-rays. Only a small change in the distribution of P3HT crystal orientations was found to be present at various levels of interdiffusion, while the intensity of the (200) peak of P3HT increased by nearly a factor of two on annealing at 170 C. It was shown that the interdiffusion process has little effect on the crystalline regions of the P3HT film, where the diffusion of PCBM into P3HT occurs within

Relation Between Nanomorphology and Performance of Polymer-Based Solar Cells 131

To determine how interdiffusion within this system affects the growth of the P3HT crystallites, the P3HT crystallite size along the *a*-axis for the bilayer films was compared to pure P3HT films heated under similar conditions (Fig. 7 (f)-(g)). The P3HT crystallite size was estimated using the Scherrer equation and plotted against the fraction of PCBM within the P3HT layer (Fig. 7 (f) ). The crystallite size was found to increase with increasing annealing temperature regardless of the level of interdiffusion. The P3HT crystallite size in the bilayer system was found to increase most rapidly during the first 5 min of annealing, where the crystallite thickness was approching that for a neat P3HT film heated under similar conditions (Fig.

Postproduction treatment requires a rather well controlled environment, it adds an additional fabrication costs to the solar cell manufacturing process, which might not be attractive for large-scale industrial production. Furthermore, some material systems, like the low band gap organic semiconductor poly[2,6-(4,4-bis-(2-ethylhexyl)-4H-cyclopenta[2,1-b;3,4-b0] dithiophene)-alt-4,7-(2,1,3-benzothiadiazole)] (PCPDTBT) blended with [6,6]-phenyl C71-butyric acid methyl ester (C71-PCBM), do not shown any improvement upon thermal

Phase separation and molecular self-organization can be influenced by solvent evaporation since the solvent establishes the film evolution environment. Slow drying or solvent annealing techniques have also been used to control the morphology of the blends by changing the rate of solvent removal.(Li et al., 2005; Li, Yao, Yang, Shrotriya, Yang & Yang, 2007; Sivula et al., 2006) The use of different solvents and their effect on the film nano-structure of BHSC has been studied in detail in the past.(Li, Shrotriya, Yao, Huang & Yang, 2007) High boiling point solvents were used with the device placed in an enclosed container, in which the atmosphere

Grazing-incidence x-ray diffraction (GIXRD) studies provided evidence that the solvent evaporation rate directly influences the polymer chain arrangement in the film.(Chu et al., 2008) It was shown that the use of higher boiling point solvent strongly improves the PCE of MDMO-PPV and PCBM blends.(Shaheen et al., 2001) Higher PCE values due to improved film morphology and crystallinity have been reached by substituting chloroform with chlorobenzene for P3HT/PCBM BHSC.(Ma et al., 2005) The difference between chlorobenzene and 1,2-dichloro benzene for use as a solvent was shown in the novel low bandgap polymer PFco-DTB and C71-PCBM blend systems, where chlorobenzene resulted in films with higher

the disordered regions of P3HT.

rapidly saturates with the solvent.

7 (g) ).

annealing.

**3.2 Solvent effects**

is visible from the increased intensity of the reflection ring in the SAED pattern (inset of Fig. 6 a)). Larger and darker PCBM rich areas can be observed suggesting an increased phase demixing between P3HT and PCBM. It was concluded that the crystallinity of P3HT is improved upon annealing and the demixing between the two components is increased, but large-scale phase separation does not occur. The resulting interpenetrating networks composed of P3HT crystals with a high aspect ratio and aggregated nanocrystalline PCBM domains provide continuous pathways in the entire photoactive layer for efficient hole and electron transport.

In order to further understand the extent of thermal annealing, 2-D X-ray scattering in a grazing incidence geometry (GIWAXS) was used to study the development of the crystalline structure of P3HT and PCBM during the interdiffusion process at various temperatures. 2D GIWAXS patterns of as-prepared P3HT/PCBM bilayers and annealed samples are shown in Fig. 7.(Treat et al., 2011) The diffraction patterns for P3HT shows that the *a*-axis of the P3HT crystals is predominantly oriented perpendicular to the substrate and the *b*-axis ( pi-stacking)

Fig. 7. Two-dimensional GIWAXS of a P3HT/PCBM bilayer on Si a) as-cast and annealed at b) 70 C, c) 110 C, d) 150 C, and e) 170 C for 5 min. f) The Scherrer equation was used to extract the P3HT crystallite thickness along the a- axis from the full-width-at-half-maximum of the (100) reflection. PCBM incorporation from the DSIMS measurements was plotted for comparison at various annealing temperatures. g) Growth in the crystal thickness with time using in-situ heating 2D GIWAXS of a P3HT/PCBM bilayer on Si at 110 C (orange) and 170 C (blue). The Scherrer equation was used to determine crystal thickness from the (100) reflection corresponding to P3HT. The dotted line corresponds to a neat P3HT/Si sample heated for 5 min at 110 C (orange) and 170 C (blue). Copyright 2011 Wiley. Used with permission from (Treat et al., 2011).

is oriented parallel - which is the typically observed P3HT orientation. Upon annealing the as-prepared films at various temperatures, the d-spacing along the *a*-axis of the P3HT crystal was found to remain constant, indicating that during the interdiffusion process, the PCBM does not interpenetrate between the side chains of the P3HT crystal structure.(Mayer et al., 2009) The peak width of the diffraction ring, corresponding to the aggregates of PCBM does not change during the interdiffusion process, showing that PCBM remains in an amorphous state with aggregates large enough to scatter incident X-rays. Only a small change in the distribution of P3HT crystal orientations was found to be present at various levels of interdiffusion, while the intensity of the (200) peak of P3HT increased by nearly a factor of two on annealing at 170 C. It was shown that the interdiffusion process has little effect on the crystalline regions of the P3HT film, where the diffusion of PCBM into P3HT occurs within the disordered regions of P3HT.

To determine how interdiffusion within this system affects the growth of the P3HT crystallites, the P3HT crystallite size along the *a*-axis for the bilayer films was compared to pure P3HT films heated under similar conditions (Fig. 7 (f)-(g)). The P3HT crystallite size was estimated using the Scherrer equation and plotted against the fraction of PCBM within the P3HT layer (Fig. 7 (f) ). The crystallite size was found to increase with increasing annealing temperature regardless of the level of interdiffusion. The P3HT crystallite size in the bilayer system was found to increase most rapidly during the first 5 min of annealing, where the crystallite thickness was approching that for a neat P3HT film heated under similar conditions (Fig. 7 (g) ).


**3. Application of deep learning**

Deep learning is new and state-of-the-art technology used for large scale applications now-days. Deep learning (also called differential programming or structure learning) is member of a large family of machine learning class. It is edge-cutting technology used for many different new research fields which are stated below.