## 1 Introduction
In [18], the parameters of the combustion process were tested with the use of the single-cylinder diesel engine with an optical combustion chamber. The test results confirmed the impact of a low start-up temperature on the parameters of the combustion process. Several studies previously focused on the simulation tests of pollutant emissions from engines under cold-start conditions [1; 9; 10] and the inventory of pollutant emissions [1; 5]. The study [10] presents an analysis of the results of the simulation tests on pollutant emissions from spark-ignition engines during cold-start, depending on: ambient temperature, the time of vehicle standstill before start-up, the length of distance covered by the vehicle after start-up and the vehicle motion model. The study [9], carried out with the use of INFRAS AG software [17] (also used in the study [10]), presents the results of the simulation tests on the emission of organic compounds from automotive combustion engines at the stage of engine warm-up after the start-up. The study [1] presents the results of simulation tests on pollutant emissions at the cold-start of spark-ignition engines for the structure of passenger cars and light trucks in Australia. This study was carried out with the use of COERT software [24]. The studies [1; 5] present the results on the inventory of pollutant emissions associated with the cold-start of spark-ignition engines. The study [5] presents the results of greenhouse gas (GHG) emissions. The study [1] examined the emission of carbon monoxide, organic compounds and nitrogen oxides, as well as fuel consumption during engine warm-up. COERT software was used for the inventory of pollutant emissions. Several studies previously addressed the methods for reducing pollutant emissions and fuel consumption during engine cold-start. In [12], test results on a compression-ignition engine with the platinum-coated glow plug were presented. Measurable effects were achieved in reducing the emissions of carbon monoxide, hydrocarbons and particulate matter, as well as fuel consumption. The large interest in pollutant emissions and fuel consumption under cold-start conditions for internal combustion engines proves that this issue has gained great interest in the area of construction and the operation of internal combustion engines. In general, total fuel consumption intensity and pollutant emission intensity can be represented by the D set consisting of two elements: the total fuel consumption intensity and the total pollutant emission intensity:  
\[\mathbf{D}=[G_{F},\mathbf{E}] \tag{1}\]  
where:  
\(G_{F}\)--total fuel consumption intensity,  
\(\mathbf{E}\)--total pollutant emission intensity--a set of which elements are the emission intensity of an individual pollutant. Additional fuel consumption and pollutant emissions are modeled, consistent with [10; 13; 17]:  
* ambient temperature--\(t_{a}\),
* rest time before starting the engine--\(\tau\),
* distance traveled by the vehicle after start-up--\(l\),
* vehicle motion conditions after engine start-up, as described by the model of vehicle movement--\(v(t)\). Additional factors of fuel consumption intensity and pollutant emission intensity are presented as a set:  
\[\mathbf{D_{w}}=[G_{Fw},\mathbf{E_{w}}] \tag{2}\]  
where:  
\(G_{Fw}\)--additional fuel consumption intensity during warm-up of the internal combustion engine, \(\mathbf{E_{w}}\)--additional pollutant emission intensity during warm-up of the internal combustion engine. Additionally, fuel consumption intensity and pollutant emission intensity are modeled as an operator [10; 13]:  
\[\mathbf{D_{w}}=\mathbf{B}[t_{a},\tau,l,v(t)] \tag{3}\]  
There are available models that enable the estimation of additional fuel consumption and pollutant emissions, among others: INFRAS AG [17] and COBERT [24] software. The emission of pollutants is most sensitive during engine warm-up, especially in the case of spark-ignition engines and--to a much lesser extent--in compression-ignition engines.