Temperature-difference driving

Heat transfer through a partition is a function of resistance to heat flow, the temperature difference driving the heat through and the surface area. The function of heating systems is to provide the heat lost in maintaining the temperature difference. Thus... 

To evaluate the true temperature difference (driving force) in a mixed vapour condenser a condensation curve (temperature vs. enthalpy diagram) must be calculated showing the change in vapour temperature versus heat transferred throughout the condenser, Figure 12.48. The temperature profile will depend on the liquid-flow pattern in the condenser. There are two limiting conditions of condensate-vapour flow ... 

The region BCD is the transition region of boiling. No active centers exist. The heat flux from the hot solid to the boiling liquid decreases continuously as the temperature-difference driving force is increased. 

Temperature difference between the condensing steam and the boiling process liquid is then (320°F — 240°F) = 80°F. This is called the temperature-difference driving force, or AT. 

Convective heat transfer results from the motion of fluids of different temperature relative to each other and is therefore intimately related to fluid flow. In general, we would expect that the convective heat flux q to be proportional to the temperature difference driving... 

During the time an evaporator is in operation, solids often deposit on the heat-transfer surfaces, forming a scale. The continuous formation of the scale causes a gradual increase in the resistance to the flow of heat and, consequently, a reduction in the rate of heat transfer and rate of evaporation if the same temperature-difference driving forces are maintained. Under these conditions, the evaporation unit must be shut down and cleaned after an optimum operation time, and the cycle is then repeated. 

If Q represents the total amount of heat transferred in the operating time Bb, and A and At represent, respectively, the heat-transfer area and temperature-difference driving force, the rate of heat transfer at any instant is... 

The instantaneous rate of heat transfer varies during the time of operation, but the heat-transfer area and the temperature-difference driving force remain essentially constant. Therefore, the total amount of heat transferred during an operating time of Bb can be determined by integrating Eq. (23) as follows ... 

At, = log-mean temperature-difference driving force over condenser, °F Hy = hours the condenser is operated per year, h/year Cw = cooling-water cost assumed as directly proportional to amount of water supplied, /lb... 

At = temperature-difference driving force (subscript lm designates log mean), °F... 

The only factor which affects the overall coefficient is the scale formation. The liquid enters the evaporator at the boiling point, and the temperature and heat of vaporization are constant. At the operating conditions, 990 Btu are required to vaporize 1 lb of water, the heat-transfer area is 400 ft2, and the temperature-difference driving force is 70°F. The time required to shut down, clean, and get back on stream is 4 h for each shutdown, and the total cost for this cleaning operation is 100 per cycle. The labor costs during operation of the evaporator are 20 per hour. Determine the total time per cycle for minimum total cost under the following conditions ... 

Derive an expression similar to Eq. (56) for finding the optimum exit temperature of cooling water from a heat exchanger when the temperature of the material being cooled is not constant. Designate the true temperature-difference driving force by... 

Many of the important cases of heat transfer involve the flow of heat from one fluid through a solid retaining wall into another fluid. This heat must flow through several resistances in series. The net rate of heat transfer can be related to the total temperature-difference driving force by employing an overall coefficient of heat transfer [/ thus, for steady-state conditions,... 

Temperature-difference driving force for counterflow based on temperature of utility fluid at exit from exchanger, At. = t - t2... 

Evaluate the mean temperature-difference driving force. 

Saturated steam is available at 50 psig. Neglect any sensible heat transfer from steam condensate to boiling liquid. The temperature-difference driving force At in the calandria may be assumed to be 90°F. 

Estimate the log-mean temperature temperature-difference driving force. The formula is LMTD = (tg — ti)/ln(tg — ti), where LMTD is log-mean temperature difference, tg is the maximum temperature difference between steam and cooling water, and is the minimum temperature difference between them. The maximum difference is (212 — 80), or 132 the minimum is (212 — 115), or 97. Accordingly, LMTD = (132 - 97)/ln( 132/97) = 113.6°F. 

This temperature drop creates temperature differences within the water at the top as well as between the water and the surrounding air. These temperature differences drive heat transfer toward the water surface from both the air and the deeper paits of the water, as shown in Figure 14-52, If the evaporation rate is high and thus the demand for the heat of vaporization is liigher than the amount of heat that can be supplied from the lower parts of the water body and the surroundings, the deficit is made up from the sensible heat of the water at the surface, and thus the temperature of water at the surface drops further. The process continues until the latent heat of vaporization equals the heat tran.sfer to the water at the surface. Once the steady operation conditions are reached and the interface temperature stabilizes, the energy balance on a thin layer of liquid at the surface can be expressed as... 

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