According to Nusselt [1], velocity and temperature profile in the condensate layer is:

$v_x(y)=\frac{g\left({\rho }_{\text{k}}-{\rho }_{\text{p}}\right)}{{\eta }_{\text{k}}}{\delta }^2\left(x\right)\left[\frac{y}{\delta \left(x\right)}-\frac{1}{2}\left(\frac{y}{\delta \left(x\right)}\right)\right]$ (1)

Velocity v_x(y) on y = δ(x) is also maximum velocity which equals:

$v_{xy}\left[y=\delta \left(x\right)\right]=v_{x,\text{maks}}=\frac{g\left({\rho }_{\text{k}}-{\rho }_{\text{p}}\right){\delta }^2\left(x\right)}{2{\eta }_k}$ (2)

$T\left(x,y\right)=T_{\text{s}}+\frac{T^{\prime }-T_{\text{s}}}{\delta \left(x\right)}y$ (3)

Eq. (1) and eq. (3) are valid within the interval:

$0\le x\le X$ (4a)

$0\le y\le \delta \left(x\right)$ (4b)

According to Nusselt [1], thickness of the resulted condensate layer is calculated using the following equation:

$\delta \left(x\right)={\left[\frac{4{\eta }_{\text{k}}{\lambda }_{\text{k}}\left(T^{\prime }-T_{\text{s}}\right)x}{gr\left({\rho }_{\text{k}}-{\rho }_{\text{p}}\right){\rho }_{\text{k}}}\right]}^{\frac{1}{4}}=C{x}^{\frac{1}{4}}$ (5a)

where constant (C) has a form:

$C={\left[\frac{4{\eta }_{\text{k}}{\lambda }_{\text{k}}\left(T^{\prime }-T_{\text{s}}\right)}{gr\left({\rho }_{\text{k}}-{\rho }_{\text{p}}\right){\rho }_{\text{k}}}\right]}^{\frac{1}{4}}$ (5b)

Eq. (1) to eq. (4) is valid as long as there is laminar condensate outflow along the vertical plane wall. According to Bejan [25], criterion which has to be fulfilled, for the case above is:

${\mathrm{Re}}_{\delta \left(x\right)}=\frac{{\overline{v}}_x\delta \left(X\right){\rho }_{\text{k}}}{{\eta }_{\text{k}}}<30$ (6)

where velocity ${\overline{v}}_X$ can be calculated from mass flow rate of condensate reduced to wall width (b):

$\frac{q_{\text{m}}}{b}=\frac{{\alpha }_{\text{m}}\left(T^{\prime }-T_{\text{s}}\right)X}{r}$ (7)

On the other side amount of value q_m/b is gained from equation:

${\Phi }_{\text{L}}={\alpha }_{\text{m}}\left(T^{\prime }-T_{\text{s}}\right)X=\frac{q_{\text{m}}}{b}r$ (8)

By inserting eq. (6) into eq. (8) followed by integration the requested value q_m/b is obtained:

$\frac{q_{\text{m}}}{b}=\frac{{\alpha }_{\text{m}}(T^{\prime }-T_{\text{s}})X}{r}$ (9)

Average convective heat transfer coefficient on a plane wall with given X is calculated according to Nusselt [1]:

${\alpha }_{\text{m}}=\frac{4}{3}\sqrt[4]{\left[\frac{{\rho }_{\text{k}}({\rho }_{\text{k}}-{\rho }_{\text{p}})gr{\lambda }_{\text{k}}^3}{4{\eta }_{\text{k}}(T΄-T_{\text{s}})X}\right]}$ (10)

From eq. (9), eq. (7) and eq. (5), velocity $[{\overline{v}}_{(X)}]$ is defined:

${\overline{v}}_x=\frac{{\alpha }_{\text{m}}\left(T^{\prime }-T_{\text{s}}\right)X}{{\rho }_{\text{k}}r\delta \left(X\right)}$ (11)

and using eq. (6), value of Nusselt number is checked, i.e., fulfilment of criteria given in eq. (6) is controlled.

Entropy production in convective heat transfer is a result of heat conduction and dissipation of mechanical energy. Their determination is relatively simple with presumed laminar condensate flow. In such laminar flow, both forms of entropy production rest on molecular level through values of thermal conductivity (λ_k) as well as dynamic viscosity (η_k), with existence of temperature gradients and velocity gradients. Based on that, the expression for local entropy production is Cengel [27]:

$S_{\text{cond}}=\frac{{\lambda }_{\text{k}}}{T^2}\left[{\left(\frac{\partial T}{\partial x}\right)}^2+{\left(\frac{\partial T}{\partial y}\right)}^2\right]$ (12)

$S_{\text{dis}}=\frac{{\eta }_{\text{k}}}{T}\left\{2\left[{\left(\frac{\partial v_x}{\partial x}\right)}^2+{\left(\frac{\partial v_y}{\partial y}\right)}^2\right]+{\left(\frac{\partial v_x}{\partial y}+\frac{\partial v_y}{\partial x}\right)}^2\right\}$ (13)

Using differential equations above, generated entropy expressed in W/mK can be expressed with the following integral equation:

${\dot{S}}_{\text{gen}}={\displaystyle {\int }_0^X{\displaystyle {\int }_0^{\delta \left(x\right)}\frac{{\lambda }_{\text{k}}}{T^2}\left[{\left(\frac{\partial T}{\partial x}\right)}^2+{\left(\frac{\partial T}{\partial y}\right)}^2\right]}}dxdy+{\displaystyle {\int }_0^X{\displaystyle {\int }_0^{\delta \left(x\right)}\frac{{\eta }_{\text{k}}}{T}}}\left\{2\left[{\left(\frac{\partial v_x}{\partial x}\right)}^2+{\left(\frac{\partial v_y}{\partial y}\right)}^2\right]+{\left(\frac{\partial v_x}{\partial y}+\frac{\partial v_y}{\partial x}\right)}^2\right\}dxdy$ (14)

Where partial derivatives of temperature with respect to x and y coordinates stand for:

$\left(\frac{\partial T}{\partial x}\right)=-\frac{T^{\prime }-T_{\text{s}}}{4C}y{x}^{-\frac{5}{4}}$ (15a)

$\left(\frac{\partial T}{\partial y}\right)=-\frac{T^{\prime }-T_{\text{s}}}{\delta \left(x\right)}$ (15b)

In observed case, the dominant temperature gradient is in y-axis direction, so in the first approximation, entropy generation for only existence of that temperature gradient is observed, which means that entropy contribution of other members in eq. (14) is ignored. So, using eq. (3) and boundary conditions in eq. (4a), eq. (4b) and eq. (14), the following equation is obtained:

${\dot{S}}_{\text{gen},\text{d}T/\text{d}y}={\lambda }_{\text{k}}{\left[\frac{T^{\prime }-T_{\text{s}}}{\delta \left(x\right)}\right]}^2{\displaystyle {\int }_0^{X}dx{\displaystyle {\int }_0^{\delta \left(x\right)}\frac{dy}{{\left[T_{\text{s}}+\frac{T^{\prime }-T_{\text{s}}}{\delta \left(x\right)}y\right]}^2}}}$ (16a)

Solution for second integral is obtained by importing the substitution:

$u=T_{\text{s}}+\frac{T^{\prime }-T_{\text{s}}}{\delta \left(x\right)}y\to y=\frac{\left(u-T_{\text{s}}\right)\delta \left(x\right)}{T^{\prime }-T_{\text{s}}};dy=\frac{\delta \left(x\right)}{T^{\prime }-T_{\text{s}}}du$ (a)

and crossing to variable (u) according to eq. (a), limits of integration are now:

$\begin{array}{cc}y=0;& u=T_{\text{s}}\end{array}$ (b)

$\begin{array}{cc}y=\delta (x);& u=T\text{'}\end{array}$ (c)

With imported values eq. (14) has a form:

${\dot{S}}_{\text{gen,d}T/\text{d}y}={\lambda }_{\text{k}}{\left(\frac{T^{\prime }-T_{\text{s}}}{\delta \left(x\right)}\right)}^2\frac{\delta \left(x\right)}{T^{\prime }-T_{\text{s}}}{\displaystyle {\int }_0^{X}dx}{\displaystyle {\int }_{T_{\text{s}}}^{T^{\prime }}\frac{du}{u^2}=\frac{{\lambda }_{\text{k}}{\left(T^{\prime }-T_{\text{s}}\right)}^2}{T_{\text{s}}{T}^{\prime }}}{\displaystyle {\int }_0^X\frac{dx}{\delta \left(x\right)}}$ (16b)

Upon conducted integration it is easy to come to the final expression for entropy generation:

${\dot{S}}_{\text{gen,d}T/\text{d}y}=\frac{4}{3}\frac{{\left(T^{\prime }-T_{\text{s}}\right)}^2}{T^{\prime }{T}_{\text{s}}}{\left[\frac{gr({\rho }_{\text{k}}-{\rho }_{\text{p}}){\rho }_{\text{k}}{\lambda }_{\text{k}}^3{X}^3}{4{\eta }_{\text{k}}\left(T^{\prime }-T_{\text{s}}\right)}\right]}^{\frac{1}{4}}$ (16c)

The following operation shows the quantitative effect of the temperature gradient in x-axis direction ∂T/∂x (as a result of heat conduction) to entropy generation:

${\dot{S}}_{\text{gen,d}T/\text{d}x}={\displaystyle {\int }_0^X{\displaystyle {\int }_0^{\delta \left(x\right)}\frac{{\lambda }_{\text{k}}}{T^2}\left[{\left(\frac{\partial T}{\partial x}\right)}^2\right]}}dxdy$ (17a)

By inserting eq. (3) and eq. (15), and boundary conditions into eq. (17a), as well as substituting with eq. (a-c), the following form of the integral is obtained:

${\dot{S}}_{\text{gen,d}T/\text{d}x}=\frac{{\lambda }_{\text{k}}\left(T^{\prime }-T_{\text{s}}\right)}{16C^2}{\displaystyle {\int }_0^X\delta \left(x\right)}{x}^{-\frac{5}{2}}{\displaystyle {\int }_{T_{\text{s}}}^{T^{\prime }}\frac{{\delta }^2\left(x\right)\left(u^2-2T_{\text{s}}u+{{\displaystyle T}}_{\text{s}}^2\right)}{u^2}}dudx$ (17b)

By returning eq. (5) into eq. (17b), the final expression for entropy generation related to conductive (molecular) heat transfer in x-axis direction is:

${\dot{S}}_{\text{gen,d}T/\text{d}x}=\frac{1}{4}{\left[\frac{{\eta }_{\text{k}}{{\displaystyle \lambda }}_{\text{k}}^5{\left(T^{\prime }-T_{\text{s}}\right)}^{5}X}{gr\left({\rho }_{\text{k}}-{\rho }_{\text{p}}\right){\rho }_{\text{k}}}\right]}^{\frac{1}{4}}\frac{{\left(T^{\prime }\right)}^2-{\left(T_{\text{s}}\right)}^2-2T_{\text{s}}{T}^{\prime }\text{In}\frac{T^{\prime }}{T_{\text{s}}}}{T^{\prime }}$ (17c)

Overall entropy generation, due to existence of temperature gradients, in both y-axis and x-axis directions, equals sum of eq. (16c) and eq. (17c), i.e.:

${\dot{S}}_{\text{gen,cond}}=\frac{4}{3}\frac{{\left(T^{\prime }-T_{\text{s}}\right)}^2}{T^{\prime }{T}_{\text{s}}}{\left[\frac{gr\left({\rho }_{\text{k}}-{\rho }_{\text{p}}\right){\rho }_{\text{k}}{{\displaystyle \lambda }}_{\text{k}}^3{X}^3}{4{\eta }_{\text{k}}\left(T^{\prime }-T_{\text{s}}\right)}\right]}^{\frac{1}{4}}+\frac{1}{4}{\left[\frac{{\eta }_{\text{k}}{\lambda }_{\text{k}}^5{\left(T^{\prime }-T_{\text{s}}\right)}^{5}X}{gr\left({\rho }_{\text{k}}-{\rho }_{\text{p}}\right){\rho }_{\text{k}}}\right]}^{\frac{1}{4}}\frac{{\left(T^{\prime }\right)}^2-{\left(T_{\text{s}}\right)}^2-2T_{\text{s}}{T}^{\prime }\text{ln}\frac{T^{\prime }}{T_{\text{s}}}}{T^{\prime }}$ (18)

The dominant temperature gradient (∂T/∂y) is in y-axis direction, which means that the entropy generation will be dominant due to that gradient, i.e., portion of entropy generation caused by existence of gradient (∂T/∂x) will be irrelevant, what is confirmed by the results of the calculation.

According to Nusselt [1], velocity profile v_x(x, y) in a condensate layer has a form:

$v_x\left(x,y\right)=\frac{g\left({\rho }_{\text{k}}-{\rho }_{\text{p}}\right)}{{\eta }_{\text{k}}}\left[\delta \left(x\right)y-\frac{1}{2}{y}^2\right]$ (19)

and therefore, along with eq. (5), scalar value of velocity gradient (v_x) in x-axis direction follows:

$\frac{\partial v_x}{\partial x}=\frac{1}{4}\frac{g\left({\rho }_{\text{k}}-{\rho }_{\text{p}}\right)}{{\eta }_{\text{k}}}Cy{x}^{-\frac{3}{4}}$ (20)

From the continuity equation:

$\frac{\partial v_x}{\partial x}+\frac{\partial v_y}{\partial y}=0$ (21)

follows the expression for scalar value of velocity gradient (v_y) in y-axis direction:

$\frac{\partial v_y}{\partial y}=-\frac{1}{4}\frac{g\left({\rho }_{\text{k}}-{\rho }_{\text{p}}\right)}{{\eta }_{\text{k}}}Cy{x}^{-\frac{3}{4}}$ (22)

By integrating the equation above along y-axis and using the condition v_y(x, y) = 0, for y = 0, the expression for velocity v_y(x, y) is obtained:

$v_y\left(x,y\right)=\frac{1}{8}\frac{g\left({\rho }_{\text{k}}-{\rho }_{\text{p}}\right)}{{\eta }_{\text{k}}}{y}^2{x}^{-\frac{3}{4}}$ (23)

From eq. (19) and eq. (23) follows:

$\frac{\partial v_x}{\partial y}=\frac{g\left({\rho }_{\text{k}}-{\rho }_{\text{p}}\right)}{{\eta }_{\text{k}}}\left(\delta -y\right)$ (24)

$\frac{\partial v_y}{\partial x}=\frac{3}{32}\frac{g\left({\rho }_{\text{k}}-{\rho }_{\text{p}}\right)}{{\eta }_{\text{k}}}{y}^2{x}^{-\frac{7}{4}}$ (25)

According to the previously developed algorithm, the results of the calculation and their interpretation are shown in this section, so the diagram in Figure 1 displays the values of convective heat transfer coefficient, heat flow rate and thickness of the resulting condensate, with regard to the height of the plane wall (X).

It can be seen from Figure 1 that the thickness of the resulting condensate δ(x), which was calculated according to eq. (5), increases with the increase in wall height (X), as does the heat flow rate (Φ_L), which is calculated according to eq. (8). It is interesting to note that the observed case revolves around extremely small thicknesses of the resulting condensate, ranging in size from 0.02 to 0.073 mm. In contrast, there is a decrease in the average convective heat transfer coefficient, calculated according to eq. (10), with an increase in wall height X.

Figure 1.

Dependence of heat transfer coefficient, heat flow rate and thickness of condensate film on the height of the plane wall X at total laminar condensation of saturated water steam with ϑ' = 100 °C and ϑ_s = 98.5 °C

It is important to observe both the local temperature profile and the local velocity profile in the resulting condensate film, whose values are determined by eq. (1) and eq. (3). Thus, the diagram in Figure 2 displays local values of temperatures ϑ [x, 0 ≤ y ≤ δ(x)], as well as local values of velocities v_x [X, 0 ≤ y ≤ δ(x)] for the three selected plane wall heights X = 100, 300 and 500 mm. It can be seen that this is a linear profile of the temperatures in the condensate film with a maximum slope of the line corresponding to a wall height of 100 mm. It is evident that all lines pass through the same ordinate values of 98.5 °C and end with different δ(x) values at the ordinate value of 100 °C. The velocity profile in the condensate film v_x [X, 0 ≤ y ≤ δ(x)], for each selected wall height X has a square parabola profile and all the curves start from the origin and end with a local maximum at y = δ(x). It can be seen that the values of local velocities v_x [X, 0 ≤ y ≤ δ(x)] increase together with the increase in X. The local maximum values are determined by eq. (2). To conclude, the highest velocity gradients are on the wall, and they disappear at y = δ(x). The diagram also shows that only low condensate outflow rates appear in the observed case of laminar condensation of dry saturated water vapour, since it is evident that the values of maximum velocities range from 0.04 to 0.09 m/s.

Figure 2

Dependence of local temperature δ [X, 0 ≤ y ≤ δ(x)] and local velocity v_x [X, 0 ≤ y ≤ δ(x)] on local condensate thickness δ(y) and wall height X = 100, 300 and 500 mm for total laminar condensation of dry saturated water steam with ϑ' = 100 °C and ϑ_s = 98.5 °C

The bibliography offers various criteria that must be satisfied in order to induce laminar condensate outflow. For example, Cengel [27] specifies the criterion of Re_δ(x) ≤ 10, while in Bejan [25] the criterion is that Re_δ(x) ≤ 30. However, Bejan [25] states that there is laminar wave flow of the condensate within this region and that said criterion can also be subject to laminar flow. However, these two criteria specify substantially different plane wall heights against which laminar outflow of the condensate occurs, which is quantitatively depicted in the diagram in Figure 3. This diagram shows that, and this is physically justified, the criterion Re_δ(x) ≤ 30 provides for significantly greater wall heights X than the criterion Re_δ(x) ≤ 10. It can also be concluded that when it comes to wall height X, its surface temperature (ϑ_s) is also important. At lower wall temperatures, the sustainability of laminar condensate outflow involves very small wall heights. The diagram displays the results of the condensation of dry saturated water vapour with the pressure of 760 mm Hg, i.e., the corresponding saturation temperature ϑ' = 100 °C. It is evident from the diagram that as the wall temperature reaches the saturation temperature, the wall height at which the laminar flow of the condensate is maintained markedly increases.

Figure 3

Dependence of wall height X on wall temperature ϑ_s and two different criterions for laminar condensate outflow with ϑ' = 100 °C

Due to the aforementioned fact, the value of ϑ_s = 98.5 °C was determined as the temperature at which the dry saturated water vapour with the temperature of 100 °C is condensed, and the following cases were calculated and analysed with regard to these conditions. Diagram in Figure 4 shows the dependence of the mass flow rate of the resulting condensate, reduced to a meter of plane wall width, depending on the wall height X. Both respective criteria Re_δ(x) ≤ 10.0, for which the value X_max = 0.44 m and Re_δ(x) ≤ 30.0, for which the value X_max = 1.90 m were included in the diagram. The diagram shows that the mass flow rate of the resulting condensate increases continuously with the increase of wall height X. It can also be concluded that small values of the mass flow rates are obtained.

The mass flow rate of the condensate is in direct relation, see eq. (7-11) with the thickness δ(x) of the resulting condensate, and therefore the diagram in Figure 5 shows the dependence of that thickness, represented by the black line, on the size X. In that same diagram, the red line represents the dependence of maximum velocity for free surface flow of the condensate on the same wall height X. The velocity is calculated according to eq. (2). It can be seen that both variables continue to grow with the increase in the height wall X. In this case too, the value of size X varied up to 1.9 m, i.e., to the upper limit of the criterion Re_δ(x) ≤ 30.0.

Figure 4.

Dependence of mass flow rate on wall height X and two different criterions for laminar condensate outflow with ϑ' = 100 °C

Figure 5.

Dependence of thickness of condensate layer and maximum velocity on wall height X for laminar condensate outflow with ϑ_s = 98.5 °C and ϑ' = 100 °C

Figure 6 presents dependence of temperature gradients ∂T/∂y and ∂T/∂x on wall height X. It is clear that dominant temperature gradient is in y-axis direction (∂T/∂y), which means that the entropy generation will be dominant due to that gradient.

The diagrams presented so far have been related to the energy and mass results of the laminar film condensation calculation. However, since the motive of the paper primarily refers to entropy analysis, the following diagrams shall relate precisely to the representation of the entropy generation regarding the described occurrence.

Figure 6.

Dependence of temperature gradients ∂T/∂y and ∂T/∂x on wall height X for laminar condensate outflow

Accordingly, the diagram in Figure 7 shows the entropy generation considering the existence of temperature gradients in the directions of the y-axis and the x-axis, respectively, i.e., towards the starting eq. (14) and towards the ultimately derived eq. (18), in which the first addend represents entropy generation resulting from the temperature gradient in the direction of the y-axis, while the second addend represents the entropy generation from the existence of a temperature gradient in the direction of the x-axis.

The diagram shows that the dominant amount of entropy generation is a result of heat conduction in the direction of the y-axis, while the entropy generation resulting from heat conduction in the direction of the x-axis is practically imperceptible. This means that the entropy generation, due to the distinct temperature gradient in the direction of the y-axis, can be considered as the total entropy generation due to the conductive heat transfer in the condensate film. The second double integral in eq. (14) describes, as already mentioned, the entropy generation resulting from the influence of condensate viscosity and the velocity gradients existing therein.

Figure 7.

Dependence of entropy generation on wall height X and temperature gradients ∂T/∂y and ∂T/∂x with ϑ_s = 98.5 °C and ϑ' = 100 °C

The diagram in Figure 8 shows the entropy generation as a result of the viscosity and sum of squares of velocity gradients. The result of the calculation is given as a function of the height of the flat wall X, while maintaining the constant sizes described in Figure 8. The diagram shows that the value of this entropy generation is continuously growing with the increase of the wall height X, in such a way that the biggest increase occurs at the beginning of the formation of the laminar film of the condensate, and later this increase gets curtailed. However, the results of the calculations show that this part of the entropy generation is extremely small and has virtually no effect on the total entropy generation.

Figure 8.

Dependence of entropy generation on wall height X and sum of squared velocity gradients (∂v_x/∂x)² + (∂v_y/∂y)² with η_k = 282 × 10^-6 Pas, ϑ_s = 98.5 °C and ϑ' = 100 °C

The diagram in Figure 9 shows the entropy generation due to the viscosity of the condensate and the square of the velocity gradient (∂v_x/∂y)². The diagram shows that this entropy input also increases with the increasing of X, in such a way that this increase is bigger at higher values of X. These results also show that the absolute amount of entropy generation is barely noticeable, yet bigger than the one from the case shown in Figure 7 for 10⁵ orders of magnitude. That is in fact in accordance with the physics of the problem, because the velocity gradient ∂v_x/∂y is by far the highest. Therefore, this entropy input has practically no meaning, so it can be ignored when calculating the total entropy balance.

Figure 9.

Dependence of entropy generation on wall height X and squared velocity gradient (∂v_x/∂y)² with η_k = 282 × 10^-6 Pas, ϑ_s = 98.5 °C and ϑ' = 100 °C

Figure 10 shows the dependence of entropy production on the flat wall height X, the square of the velocity gradient (∂v_y/∂x)², and the viscosity of the condensate. This diagram also shows that this is also a very small, therefore irrelevant, entropy input. Its greatest value is at the formation of a laminar condensate film and upon reaching the wall height of 0.4 m it gets to zero value.

Figure 10.

Dependence of entropy generation on wall height X and squared velocity gradient (∂v_y/∂x)² with η_k = 282 × 10^-6 Pas, ϑ_s = 98.5 °C and ϑ' = 100 °C

Finally, the diagram in Figure 11 shows the entropy generation depending on the magnitude of X, the viscosity of the condensate and the product of the velocity gradient 2(∂v_x/∂y) (∂v_y/∂x). The result is similar in flow and values to the result given in Figure 8, which means that this part of the entropy generation can also be ignored.

Figure 11.

Dependence of entropy generation on wall height X and product of velocity gradients 2(∂v_x/∂y) (∂v_y/∂x) with η_k = 282 × 10^-6 Pas, ϑ_s = 98.5 °C and ϑ' = 100 °C

Considering the viscosity and velocity gradients during laminar condensation of vapour, the total entropy production is obtained simply by summing the values shown in Figures 8-11. Of course, this total would be extremely small, only 10^-7 W/mK, when it comes to the order of magnitude, so this total amount may be ignored with respect to the entropy generation deduced by means of eq. (16c).

The second substance chosen to be completely condensed on a vertical flat wall with laminar condensate outflow is dry saturated vapour of pure ammonia with the saturation temperature ϑ' = 100 °C and the wall temperature of 98.5 °C (exactly the same as in the previous example). The saturation pressure was 62.553 bar and the following diagrams show some of the calculation results.

Henceforth, the diagram in Figure 12 shows the dependence on the flat wall height X, convective heat transfer coefficient and heat flow rate during the complete condensation of the ammonia vapour. The behaviour of these two values is similar to the one shown in Figure 1, except that in the latter case the values of convective heat transfer coefficient and heat flow rate are lower.

Figure 12.

Dependence of heat transfer coefficient and heat flow rate on plane wall height X at condensation of saturated ammonia steam with ϑ' = 100 °C and ϑ_s = 98.5 °C

The diagram in Figure 13 shows the results of the calculation of the mass flow of the resulting condensate as well as its thickness, once again depending on the magnitude of X. It can be seen that in this case both values continuously grow with the increase in X, and in this case, too, only low values of thickness of the condensate film δ(x) and mass flow are produced.

Figure 13.

Dependence of condensate thickness and mass flow rate on wall height X for laminar ammonia condensate outflow with ϑ_s = 98.5 °C and ϑ' = 100 °C

Finally, the diagram in Figure 14 shows the entropy generation solely because of the existence of temperature gradients in the ammonia condensate stream, depending on the size of the variable X. The flow of these curves is similar to the flow of the curves given in Figure 7, with the difference being that lower condensation of ammonia vapour produces lower entropy generation values. The case of ammonia condensation also shows that the overall entropy generation is related exclusively to the temperature gradient ∂T/∂y, as is the case with condensation of water vapour. This case also proves that the entropy generation resulting from the temperature gradient of ∂T/∂x has the order of magnitude amounting to 10^-11 W/mK.

Figure 14.

Dependence of entropy generation on wall height X and temperature gradients ∂T/∂y and ∂T/∂x in ammonia condensate with ϑ_s = 98.5 °C and ϑ' = 100 °C

The results of the calculation of the entropy generation resulting from the viscosity and the existence of velocity gradients have also proven to be imperceptible, almost equal to zero, so the results are not presented in this paper.